+382 1. It is known that a fair coin is flipped 10 times. Given that the coin landed heads up at least 3 times, what is the odds that it landed heads up 7 times?
2. Given that the members of set M are all the positive factors of 25 and the members of set D are all the positive factors of 20, what is the odds that a number from set M is also in set D?
+6251 \(\text{# of heads out of 10 is binomially distributed }n=10, ~p=\dfrac 1 2\\ A \sim \{\text{7 heads}\}\\ B \sim \{\text{at least 3 heads}\}\\ P[A|B] = \dfrac{P[B|A]P[A]}{P[B]} \)
\(P[B|A] = 1\\ P[A]=\dbinom{10}{7} 2^{-10} = \dfrac{15}{128}\\ P[B] = 1-\left(\sum \limits_{k=0}^2~\dbinom{10}{k}2^{-10}\right) = \dfrac{121}{128}\)
\(P[A|B] = \dfrac{1 \cdot \frac{15}{128}}{\frac{121}{128}} = \dfrac{15}{121}\)
\(\text{odds} = \dfrac{P[A|B]}{1-P[A|B]} = \dfrac{15}{121-15} = \dfrac{15}{106}\)
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