1. zachery divides a deck of cards into two piles, each with 26 cards. he examines both piles and counts…

1. zachery divides a deck of cards into two piles, each with 26 cards. he examines both piles and counts that there are 10 spades in the first pile, and there are only 3 spades in the second pile. he asks his friend jeremy to randomly pick a pile. jeremy then draws a card from the pile he picked and reveals that the card is a spade. based on this observation, what is the probability that jeremy picked the pile with only 3 spades in it?

1. zachery divides a deck of cards into two piles, each with 26 cards. he examines both piles and counts that there are 10 spades in the first pile, and there are only 3 spades in the second pile. he asks his friend jeremy to randomly pick a pile. jeremy then draws a card from the pile he picked and reveals that the card is a spade. based on this observation, what is the probability that jeremy picked the pile with only 3 spades in it?

Answer

Explanation:

Step1: Define events

Let $A$ be the event that Jeremy picked the pile with 3 spades, and $B$ be the event that a spade is drawn.

Step2: Calculate prior probabilities

The probability of picking either pile is $P(A)=P(\overline{A})=\frac{1}{2}$.

Step3: Calculate conditional probabilities

The probability of drawing a spade from the pile with 3 spades is $P(B|A)=\frac{3}{26}$, and from the pile with 10 spades is $P(B|\overline{A})=\frac{10}{26}$.

Step4: Apply Bayes' theorem

By Bayes' theorem $P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|\overline{A})P(\overline{A})}$. Substitute the values: [ \begin{align*} P(A|B)&=\frac{\frac{3}{26}\times\frac{1}{2}}{\frac{3}{26}\times\frac{1}{2}+\frac{10}{26}\times\frac{1}{2}}\ &=\frac{\frac{3}{52}}{\frac{3 + 10}{52}}\ &=\frac{3}{3 + 10}\ &=\frac{3}{13} \end{align*} ]

Answer:

$\frac{3}{13}$