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

a) 0.5

b) 1

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) = (1)/(3)\\\\P(E) = (2)/(3)`

`P(C_2) = (1)/(3)`

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

`P(C_2|E)\\\\=(P(C_2)\cap P(E))/(P(E))\\\\=((1)/(3))/((2)/(3))=(1)/(2)`

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

`P(E|C_2)\\\\=(P(E)\cap P(C_2))/(P(C_2))\\\\= ((1)/(3))/((1)/(3)) = 1`