a couple is planning to have seven children. what is the probability that at least 3 of the children will have a birthday in the same month
To calculate the probability that at least 3 children will have a birthday in the same month, we can use the concept of probability and combinatorics. Let's break down the problem step by step:
Step 1: Determine the total number of possible outcomes.
In this case, each of the seven children can have a birthday in any of the twelve months. Therefore, the total number of possible outcomes is 12^7 because each child has 12 possible choices. Thus, there are 12^7 = 35,831,808 total outcomes.
Step 2: Determine the number of favorable outcomes.
To find the number of favorable outcomes, we need to count the number of ways in which at least 3 children will have a birthday in the same month. We can consider the cases of having exactly 3, 4, 5, 6, or 7 children with the same birth month separately.
Case 1: Exactly 3 children with the same birth month:
There are 12 months to choose from for the 3 children, and we need to select 3 children from the 7 total children. Hence, there are C(7, 3) = 35 ways to choose 3 children and C(12, 1) = 12 ways to select a month. Therefore, the number of favorable outcomes for this case is 35 × 12 = 420.
Case 2: Exactly 4 children with the same birth month:
Similarly, there are C(7, 4) = 35 ways to choose 4 children and C(12, 1) = 12 ways to select a month. The number of favorable outcomes for this case is 35 × 12 = 420.
Case 3: Exactly 5 children with the same birth month:
Following the same principle, there are C(7, 5) = 21 ways to choose 5 children and C(12, 1) = 12 ways to select a month. The number of favorable outcomes for this case is 21 × 12 = 252.
Case 4: Exactly 6 children with the same birth month:
In this case, there are C(7, 6) = 7 ways to choose 6 children and C(12, 1) = 12 ways to select a month. The number of favorable outcomes for this case is 7 × 12 = 84.
Case 5: All 7 children with the same birth month:
There is only 1 way for all 7 children to have the same birth month, and we have C(12, 1) = 12 ways to select that month. Therefore, the number of favorable outcomes for this case is 1 × 12 = 12.
Step 3: Sum up the favorable outcomes.
To find the total number of favorable outcomes, we need to sum up the favorable outcomes from each case:
Total favorable outcomes = 420 + 420 + 252 + 84 + 12 = 1188.
Step 4: Calculate the probability.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes = 1188 / 35,831,808.
Therefore, the probability that at least 3 children will have a birthday in the same month is approximately 0.00003314781 or 0.00331% (rounded to five decimal places).