11. A couple have 3 children. None of the children were born in a leap year. The
probability that all three children share the same birthday is:
a. 3/365
b. (1/365)2
c. (1/365)3
d. (1/365) x (1/364) x (1/363)
Correct Answer
The correct answer is option b. (1/365)2. The probability that all three children share the same birthday (same day and month) in a non-leap year is
1 / 365^2.
Step-by-Step Solution
There are 365 possible birthdays in a non‑leap year (ignoring the year of birth and assuming all days are equally likely).
- The birthday of the first child can be any day of the year, so no restriction or probability factor is needed for the first child.
- The second child must have exactly the same birthday as the first child. The probability of this is
1/365. - The third child must also have exactly the same birthday as the first child. The probability of this is again
1/365.
Using the multiplication rule for independent events, the probability that both the second and third child match the first child is:
P(all 3 share same birthday)
= (1/365) × (1/365)
= 1 / 365²
Therefore, the required probability is (1/365)2.
Explanation of Each Option
| Option | Expression | Meaning | Correct? | Reason |
|---|---|---|---|---|
| a | 3/365 | As if “any one of three” matches some fixed day. | No | This is linear in 1/365 and does not use the multiplication rule for joint probability of all three matching. |
| b | (1/365)2 | Second matches first, third matches first, as independent events. | Yes | Correct use of independence; probability that both the second and third child match the first child is (1/365) × (1/365) = 1/365². |
| c | (1/365)3 | All three matching some pre‑chosen specific calendar date. | No | This would be correct only if the question specified a particular date in advance (for example, all three are born on 1st January). |
| d | (1/365) × (1/364) × (1/363) | Pattern used for “all different birthdays”. | No | This structure corresponds to the probability that no two share a birthday, not that all three have identical birthdays. |
