G Assuming that P(x) is a Vector Space, prove that for all positive integer n

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Azihan

23.05.2022

Mathematics

Let A_i be the event that there are no birthdays in the $i_{th}$ season. The probability that all seasons occur at least once is

$1-Pr(A_1\cup A_2\cup A_3\cup A_4).$

Note that

$A_1\cap A_2\cap A_3\cap A_4=\emptyset.$

Using the inclusion-exclusion principle and the symmetry of the seasons,

$Pr(A_1\cup A_2\cup A_3\cup A_4)=\sum \limits_{i=1}^4Pr(A_i)-\sum \limits_{i=1}^3\sum \limits_{ji}Pr(A_i\cap A_j)+\\ \\+\sum \limits_{i=1}^3\sum \limits_{ji}\sum \limits_{kj}Pr(A_i\cap A_j\cap A_k)=4Pr(A_1)-6Pr(A_1\cap A_2)+4Pr(A_1\cap A_2\cap A_3).$

Now find:

$Pr(A_1)=\left(\dfrac{3}{4}\right)^7;$ $Pr(A_1\cap A_2)=\left(\dfrac{2}{4}\right)^7;$ $Pr(A_1\cap A_2\cap A_3)=\left(\dfrac{1}{4}\right)^7.$

Thus,

$Pr(A_1\cup A_2\cup A_3\cup A_4)=4Pr(A_1)-6Pr(A_1\cap A_2)+4Pr(A_1\cap A_2\cap A_3)=4\cdot \left(\dfrac{3}{4}\right)^7-6\cdot \left(\dfrac{2}{4}\right)^7+4\cdot \left(\dfrac{1}{4}\right)^7=\dfrac{4\cdot 3^7-6\cdot 2^7+4}{4^7}.$

The probability that all 4 seasons occur at least once is

$1-\dfrac{4\cdot 3^7-6\cdot 2^7+4}{4^7}=\dfrac{4^7-4\cdot 3^7+6\cdot 2^7-4}{4^7}\approx 0.51269... \approx 0.51$

0.51.

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