_{1}

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Short-term memory allows individuals to recall stimuli, such as numbers or words, for several seconds to several minutes without rehearsal. Although the capacity of short-term memory is considered to be 7 ± 2 items, this can be increased through a process called chunking. For example, in Japan, 11-digit cellular phone numbers and 10-digit toll free numbers are chunked into three groups of three or four digits: 090-XXXX-XXXX and 0120-XXX-XXX, respectively. We use probability theory to predict that the most effective chunking involves groups of three or four items, such as in phone numbers. However, a 16-digit credit card number exceeds the capacity of short-term memory, even when chunked into groups of four digits, such as XXXX-XXXX-XXXX-XXXX. Based on these data, 16-digit credit card numbers should be sufficient for security purposes.

Short-term memory allows stimuli, such as numbers or words, to be recalled for several seconds to several minutes without rehearsal. Miller (1956) reported that the storage capacity of short-term memory was 7 ± 2 items, naming this “the magical number” [

In probability theory, there is a problem entitled “the tourist with a short memory” [

When a subject responds to an event involving several stimuli, those stimuli must be processed in such a way to distinguish among them while still associating them with the entire set of items. According to Miller’s and Cowan’s hypotheses (7 ± 2 or 4 - 5 items, respectively) [

Assumption 1: Input items (or stimuli) are assumed to be labeled as A_{1}, A_{2}, …, and A_{n} in order.

Assumption 2: Items are remembered equally with no one item being more dominant. The probability to recall any A_{j} except A_{i} next after A_{i} is recalled is equal.

Assumption 3: The subject can only recall n items in order after he recalls every item at least once.

Applying these assumptions to the problem of “the tourist with a short memory”, the problem is to find the expected number, E(N), of trips required until the tourist has visited all capitals. The process that any A_{j} except A_{i} is recalled after A_{i} is represented as a way: W(A_{i}→A_{j}). This can be calculated without chunking (Procedure 1) and with chunking into same-sized or different-sized groups (Procedures 2 and 3) as follows:

Procedure 1: To find the expected number, E(N), of ways required until all A_{i}’s are recalled (

Procedure 2: Items are chunked in order into groups, which have the same number of items (_{1}, A_{2}, A_{3}), (A_{4}, A_{5}, A_{6}), (A_{7}, A_{8}, A_{9}), …, and (A_{n}_{−}_{2}, A_{n}_{−1}, A_{n}). Groups are denoted in order as B_{1}, B_{2}, B_{3}, … (_{j} except B_{i} immediately after B_{i} is recalled. When any B_{j} is recalled for the first time, all items in B_{j} are recalled at least once, which assumes that the relationship among the items in B_{j} has already been confirmed. Hence, all visits within B_{j} are remembered from the second visit of B_{j} onwards. When all B_{i}’s are recalled, all A_{i}’s are also recalled, confirming the relationship among all A_{i}’s.

Procedure 3: Items are chunked into groups with different numbers of items. For example, in Japan, 11-digit cellular phone numbers and 10-digit toll free numbers are displayed as 090-XXXX-XXXX and 0120-XXX- XXX, respectively. The 11-digit phone number is chunked into three groups, B_{1}, B_{2}, B_{3}, one of which consists of three digits, B_{1} = (A_{1}, A_{2}, A_{3}), and two of which consist of four digits, B_{2} = (A_{4}, A_{5}, A_{6}, A_{7}), B_{3} = (A_{8}, A_{9}, A_{10}, A_{11}). Similarly, the 10-digit toll free number is chunked into three groups, B_{1}, B_{2}, B_{3}, one of which consists of four

Ways, W(A_{i}→A_{j}), (labeled by turns) required until all A_{i}’s are recalled without any chunking of items (a) and with chunking of items into, for example, four groups (B_{1}, B_{2}, B_{3}, B_{4}) (b)

digits, B_{1} = (A_{1}, A_{2}, A_{3}, A_{4}), and two of which consist of three digits, B_{2} = (A_{5}, A_{6}, A_{7}), B_{3} = (A_{8}, A_{9}, A_{10}).

When the number of all items is n, the expected number, E(N), of ways, W(A_{i}→A_{j}), required until all A_{i}’s are recalled can be calculated.

In the case of n = 3, a subject wants to recall three items, A_{1}, A_{2}, A_{3}.

Set N as follows:

where Y_{m} is the number of ways required for recalling one more item when m items have already been recalled. Therefore, Y_{m}’s are independent stochastic variables. Y_{0} and Y_{1} are always 1. Y_{0} = 1 indicates the first way of recalling one of the items. For example, it corresponds to the first way of _{2}, one item has yet not been recalled, but it is recalled the k^{th} time with a geometric probability of

for k = 1, 2, …. The expected distribution is 1/p. Therefore, E(Y_{2}) = 2. Since Y_{m}’s are mutually independent random variables,

This equation is transformed into

When Y_{2} = y_{2}, the probability of N; P(N: Y_{2} = y_{2}), is expressed as

P(Y_{0} =1) and P(Y_{1} = 1) are always 1.

Therefore,

As

Hence,

In the case of n = 4, a subject wants to recall four items, A_{1}, A_{2}, A_{3}, A_{4}.

Set N as follows:

where Y_{i} is the number of ways required for recalling one more item when i items have already been recalled. Therefore, Y_{i}’s are mutually independent random variables. Y_{0} and Y_{1} are always 1. In the case of Y_{2}, two items have not yet been recalled, so one of these two items is recalled the k^{th} time with a geometric probability of

for k = 1, 2, …. Similarly, _{2}) = 3/2 and E(Y_{3}) = 3. Since Y_{i}’s are mutually independent random variables,

This equation is transformed into

Therefore, the expression for a general number, n, of items is:

This can be easily proven.

When Y_{2} = y_{2} and Y_{3} = y_{3}, the probability of N; P(N: Y_{2} = y_{2}, Y_{3} =y_{3}), is expressed as

P(Y_{0} = 1) and P(Y_{1} = 1) are always 1.

Therefore,

As

Hence,

In the case of n = 5,

For a general number, n (≥3), of items,

The equation is proved (Appendix). Specifically, in the case of n = 2, E(N) = 2 with a probability of 1; in the

case of n = 3, E(N) = 4, and the cumulative probability that N is smaller than or equal to E(N),

is 0.75; in the case of n = 4, E(N) = 13/2 and

In the case of n = 5, which corresponds to one of Miller’s magical numbers,

cumulative probability of

Items are chunked in order into groups with all groups containing the same number of items. The number of all items is denoted as n, and the number of items in each group is denoted as m. For an example of m = 3, the groups are (A_{1}, A_{2}, A_{3}), (A_{4}, A_{5}, A_{6}), (A_{7}, A_{8}, A_{9}), … (A_{n}_{−}_{2}, A_{n}_{−}_{1}, A_{n}). These groups are denoted in order as B_{1}, B_{2}, B_{3}, … (_{j} except B_{i} immediately after B_{i}. When any B_{j} is recalled for the first time, all items in B_{j} are recalled at least once, so it is assumed that the relationship among the items in B_{j} has already been confirmed. Hence, all visits within B_{j} are saved from the second visit of B_{j} onwards. When all B_{i}’s are recalled, it means that all A_{i}’s are recalled, confirming the relationship among all A_{i}’s.

(a) The expectation of the number, E(N), of ways, W(A_{i}→A_{j}), required until all A_{i}’s are recalled; (b) The cumulative probability that N is smaller than or equal to E(N), P(N ≤ E(N)). n represents the number of items

The number of B_{i}’s is

The expected number, E(N_{n}_{,m}), of ways required until all A_{i}’s are recalled can be calculated. For the example of n = 12 and m = 3, a subject wants to recall 12 items, A_{1}, A_{2}, A_{3}, … ,A_{12}. Then, B_{1} = (A_{1}, A_{2}, A_{3}), B_{2} = (A_{4}, A_{5}, A_{6}), B_{3} = (A_{7}, A_{8}, A_{9}), and B_{4} = (A_{10}, A_{11}, A_{12}).

Set N_{12,3} as follows:

where Z_{i} is the number of ways required for recalling one more group when i groups have been recalled, and Y_{j} is the number of ways required for recalling one more item of any group when j items of this group have been recalled. Therefore, Z_{i}’s and Y_{j}’s are mutually independent random variables. Z_{0}, Z_{1}, and Y_{1} are always 1. Specifically, Z_{0} = 1 indicates the first way going to one of the groups. For example, it corresponds to the first way of

Using the case of four items in Procedure 1, we can regard the four groups in Procedure 2 as four items,

Using the case of three items from Procedure 1,

Hence,

As another practical example, the expected number of ways required to recall 16 digits, E(N_{16,4}), corresponding to a credit card account number, XXXX-XXXX-XXXX-XXXX, can be calculated.

Using the case of four items in Procedure 1 and regarding the four groups as four items,

Using the case of four items in Procedure 1,

Hence,

Generally,

Then, if _{n}_{,m}) ca-

nonly be calculated precisely when n is a multiple of m. However, even if n is not a multiple of m, E(N_{n}_{,m}) is calculated to observe the relationship between m and E(N_{n}_{,m}). This calculation will be justified when n is larger than m, for example _{n}_{,m}) is calculated only when n is a multiple of m. E(N_{n}_{,m}) is calculated for n = 10, 11, …, 100, and m = 1, 2, …, 10. _{n}_{,m}) is the smallest and the second smallest for any_{n}_{,m}) is the third smallest. It is interesting to note that the case of m = 1 corresponds to any case without chunking from Procedure 1.

The expected number E(N_{n}_{,*}) of ways required until all A_{i}’s are recalled can be calculated in the same manner as Procedure 2 for special cases of items chunked into groups of different lengths. When lengths of chunked groups, m = 2, 3, or 4, E(N_{n}_{,m}) is the smallest. All integers are expressed by a sum of 2’s, 3’s, and 4’s. For example,

The 11-digit phone number 090-XXXX-XXXX is chunked into three groups, B_{1}, B_{2}, B_{3}, one of which consists of three digits, B_{1} = (A_{1}, A_{2}, A_{3}), and two of which consist of four digits, B_{2} = (A_{4}, A_{5}, A_{6}, A_{7}), B_{3} = (A_{8}, A_{9}, A_{10}, A_{11}).

Hence,

The 10-digit phone number 0120-XXX-XXX is chunked into three groups, B_{1}, B_{2}, B_{3}, one of which consists of four digits, B_{1} = (A_{1}, A_{2}, A_{3}, A_{4}), and two of which consist of three digits, B_{2} = (A_{5}, A_{6}, A_{7}), B_{3} = (A_{8}, A_{9}, A_{10}).

Hence,

A 10-digit phone number of 03-XXXX-XXXX (for example, in Tokyo) is chunked into three groups, B_{1}, B_{2}, B_{3}, one of which consists of two digits, B_{1} = (A_{1}, A_{2}), and two of which consist of four digits, B_{2} = (A_{3}, A_{4}, A_{5}, A_{6}), B_{3} = (A_{7}, A_{8}, A_{9}, A_{10}).

Hence,

As the number of the items, n, increases, the expected number, E(N), of ways required until all items are recalled

The expected number, E(N_{n,m}), of ways required until all A_{i}’s are recalled. n represents the number of items. m represents the number of chunked groups

increases exponentially. The cumulative probability that N is smaller than or equal to E(N), P(N ≤ E(N)), decreases steadily. Hence, the greater the number, n, of items, the greater the difficulty to recall all items. In the case of five items, which corresponds to one of Miller’s magical numbers (7 − 2 = 5), E(N) = 28/3 ≒ 10, and in the case of nine items, which corresponds to the other of Miller’s magical numbers (7 + 2 = 9), E(N) = 796/35 ≒ 23. In the case of n = 10, E(N) = 7409/280 ≒ 27.

E(N_{n}_{,m}) is the expected number of ways required until all items are recalled. Hence, a smaller value for E(N_{n,m}) indicates more efficient recall. For example, the expected number of ways required until 12 items chunked into three groups are recalled, E(N_{12,3}), is 37/2 ≒ 19. In the case of a 16-digit credit card number, XXXX-XXXX- XXXX-XXXX, E(N_{16,4}) = 57/2 ≒ 29. From the results for n = 10, 11, …, 100, and m = 1, 2, …, 10, E(N_{n,m}) is the smallest for any n (10 ≤ n ≤ 100), when m = 3 or 4. Hence, when m = 3 or 4, all items can be recalled most quickly.

The expected number of ways required to recall all 11 digits (e.g., in the phone number 090-XXXX-XXXX), E(N_{11,*}), is 18. For a 10-digit phone number in the format 0120-XXX-XXX, E(N_{10,*}) = 31/2 ≒ 16. For a 10-digit phone number in the format 03-XXXX-XXXX, E(N_{10,*}) = 16.

Short-term memory lasts from several seconds to several minutes. Based on the current data, we conclude that an individual can follow the 23 ways required to recall nine items within several minutes, but it takes longer to follow the 27 ways required to recall 10 items, so some one of the items are forgotten. These results suggest that 23 ways may be the critical number, beyond which some items will be forgotten.

A smaller number of E(N_{n}_{,m}) indicates more efficient recall. From the results for n = 10, 11, …, 100, and m = 1, 2, …, 10, E(N_{n}_{,m}) is the smallest for any n, (10 ≤ n ≤ 100) when m = 3 or 4. Each group has 3 or 4 items (m = 3 or 4) without chunking. From Procedure 1, P(N ≤ E(N)) is 0.75 in the case of three items, and P(N ≤ E(N)) is 0.7407 in the case of four items. P(N ≤ E(N)) decreases steadily with more items. Hence, when m = 3 or 4, all items of each group can be recalled most quickly and with the greatest confidence. E(N_{12},_{3}) = 37/2 ≒ 19 is less than 23, the critical number for recall. Hence, chunking will be effective: B_{1} = (A_{1}, A_{2}, A_{3}), B_{2} = (A_{4}, A_{5}, A_{6}), B_{3} = (A_{7}, A_{8}, A_{9}), B_{4} = (A_{10}, A_{11}, A_{12}). However, for 16 digits, such as in a credit card number, XXXX-XXXX- XXXX-XXXX, E(N_{16,4}) = 57/2 ≒ 29, which is larger than the critical number for recall. Thus chunking will not benefit short-term memory recall of a 16-digit credit card number. Based on these findings, a 16-digit credit card number of XXXX-XXXX-XXXX-XXXX should have greater security than a 12-digit number of XXX-XXX- XXX-XXX.

The expected numbers, E(N), of ways for 090-XXXX-XXXX, 0120-XXX-XXX, and 03-XXXX-XXXX, are less than 23, the critical number for recall. Hence, chunking into groups of two to four items is truly effective for recalling 11 or 10-digit phone numbers.

The current findings were obtained using a model based on certain assumptions. The validity of these assumptions should be investigated in the future.

We use probability theory to predict that the most effective chunking involves groups of three or four items, such as in phone numbers, and conclude that an individual can follow the 23 ways required to recall nine items within several minutes, but it takes longer to follow the 27 ways required to recall 10 items, so some of the items are forgotten. These results suggest that 23 ways may be the critical number, beyond which some items will be forgotten. A 16-digit credit card number exceeds the capacity of short-term memory, even when chunked into groups of four digits, such as XXXX-XXXX-XXXX-XXXX. Based on these data, 16-digit credit card numbers should be sufficient for security purposes.