Question

You have a shuffled deck of three cards: 2, 3, 4. You draw one card. Let C_i denote the event that the card i is picked. Let E denote the event that the card chosen is an even-numbered card. a. What is P[C_2|E], the probability that the 2 is chosen given that an even-numbered card is chosen? b. What is the conditional probability that an even-numbered card is picked given that a 2 is picked?

Answer 1

Answer:

a) 0.5

b) 1  

Step-by-step explanation:

We are given the following in the question:

We have a deck of 3 cards: 2, 3, 4.

$C_i$: card i is picked.

E: card chosen is an even-numbered card.

Definition:

$\text{Probability} = \displaystyle\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$

$P(C_i) = \dfrac{1}{3}\\\\P(E) = \dfrac{2}{3}$

$P(C_2) = \dfrac{1}{3}$

a) probability that the 2 is chosen given that an even-numbered card is chosen

$P(C_2|E)\\\\=\dfrac{P(C_2)\cap P(E)}{P(E)}\\\\=\dfrac{\frac{1}{3}}{\frac{2}{3}}=\dfrac{1}{2}$

b) conditional probability that an even-numbered card is picked given that a 2 is picked

$P(E|C_2)\\\\=\dfrac{P(E)\cap P(C_2)}{P(C_2)}\\\\= \dfrac{\frac{1}{3}}{\frac{1}{3}} = 1$