Computer Science · Oct / Nov 2024 · Paper 2 HL · Question 5

5. The probability of two people having the same birthday is 1/365 or 0.003. The probability of two people (one pair) from a random sample of N people sharing a birthday is calculated as follows: y Number of pairs (NP): (N*(N-1))/2 y Probability of a pair with different birthdays (PP): 364/365 = 0.997 y Probability of different pairs (DP) for a specified number of pairs (NP): PP^NP y Probability of a pair sharing a birthday (SB): 1-DP Table 1 shows how the probability of two people sharing a birthday is calculated in a sample of 23 people. Table 1: Calculating probabilities of pairs sharing a birthday Formula Result Number of pairs (NP) (N*(N-1))/2 = (23*22) / 2 253 Different pairs (DP) PP^NP = 0.997^253 0.500 Pair sharing a birthday (SB) 1-DP 0.500 Figure 2 shows the probability of two people sharing the same birthday, where N is the number of people in the sample. Figure 2: Probability of two people sharing the same birthday 0 0.00 0.25 0.50 0.75 1.00 10 20 30 40 Number of people in sample Probability of a shared birthday (Option B continues on the following page) (Option B, question 5 continued) The graph in Figure 2 was created from the spreadsheet shown below with the values shown in the first two rows: N NP DP SB 2 1 0.997 0.003 3 3 0.995 0.005

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