Probability that All Four Seasons Occur Among 7 Birthdays

A group contains 7 people. Each person’s birthday is equally likely to occur in any of the 4 seasons: Spring, Summer, Autumn, Winter. We find the probability that all four seasons occur at least once.

Total Sample Space

Each person has 4 choices, so total outcomes are:

$4^7=16384$

Favourable Outcomes (Inclusion–Exclusion)

We subtract the cases where one or more seasons are missing.

Number of favourable outcomes:

$4^7-\binom{4}{1}3^7+\binom{4}{2}2^7-\binom{4}{3}1^7$

Substituting values step-by-step:

$4^7=16384$
$3^7=2187$
$2^7=128$
$\binom{4}{1}=4$
$\binom{4}{2}=6$

$\binom{4}{3}=4$

Now compute:

$16384-4(2187)+6(128)-4(1)$
$=16384-8748+768-4$

$=8400$

So the number of favourable assignments is 8400.

Probability

$P=\frac{8400}{4^7}=\frac{8400}{16384}$

Simplify:

$P=\frac{525}{1024}\approx0.5127$

So the probability is approximately 51.27%.

Distribution Method (Conceptual Check)

To ensure all four seasons occur, the only possible distributions of 7 people are:

(4,1,1,1), (3,2,1,1), (2,2,2,1)

Corresponding number of arrangements:

$\frac{7!}{4!1!1!1!}=210$ $4\times210=840$ $\frac{7!}{3!2!1!1!}=420$ $4\times3\times420=5040$ $\frac{7!}{2!2!2!1!}=630$ $4\times630=2520$

Total favourable outcomes:

$840+5040+2520=8400$

This confirms the earlier result.

Final Answer

$P=\frac{525}{1024}\approx51.27%$

There is about a 51.27% chance that all four seasons occur at least once among the birthdays of 7 people.