Question

Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deck of 52 cards without replacement. 1) What’s the probability that both of the cards are hearts? 2) What’s the probability that exactly one of the cards are hearts? 3) What’s the probability that none of the cards are hearts? Make sure to clearly define your probabilistic events and mathematically show how different probability laws and rules that you learned in class could be applied to solve the problems.

Answer 1

0.0588235; 0.3823529; 0.5588235

Explanation:

Given that:

Total number of cards = 52

Total number of hearts = 13

Probability = required outcome / Total possible outcomes

Two cards are drawn without replacement :

1) What’s the probability that both of the cards are hearts?

First draw :

P(hearts 1) = 13 / 52

Second draw :

P(hearts 2) = 12 / 51

P(both cards being hearts) :

(13 /52) × (12/51) = 156 / 2652 = 26/442 = 13 / 221 = 0.0588235

2) What’s the probability that exactly one of the cards are hearts?

First draw being hearts, 2nd not hearts Or First draw not hearts and 2nd being hearts

P(first draw hearts) = 13 /52

P(2nd not hearts) = 39/51

13/52 × 39/51 = 0.1911764

P(first not hearts) = 39 /52

P(2nd being hearts) = 13 / 51

39/52 × 13/51 = 0.1911764

Hence, (0.1911764 + 0.1911764) = 0.3823529

3) What’s the probability that none of the cards are hearts

P(first draw, no hearts) = 39/52

P(second draw, no hearts) = 38/ 51

39/52 × 38/51 = 1482 / 2652 = 0.5588235