In a Certain Game, You Pick a Card From a Standard Deck GMAT Problem Solving

Question: In a certain game, you pick a card from a standard deck of 52 cards. If the card is a heart, you win. If the card is not a heart, the person replaces the card to the deck, reshuffles, and draws again. The person keeps repeating that process until he picks a heart, and the point is to measure how many draws it took before the person picked a heart and, thereby, won. What is the probability that one will pick the first heart on the third draw or later?

1. 1/2
2. 9/16
3. 11/16
4. 13/16
5. 15/16

‘In a Certain Game, You Pick a Card From a Standard Deck GMAT Problem Solving’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”. To understand GMAT questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude for reasoning and mathematics. The GMAT Quantitative Test consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

Solution and Explanation:
Approach Solution 1:
Favorable case = the heart is picked in the third draw or later
Unfavorable case = The heart is picked in either first draw or in second draw
Probability = Favorable outcomes / Total out comes
Also probability = 1-(Unfavorable outcomes / Total out comes)
Unfavorable case 1: probability of heart picked in first draw = 13/52 =1/4
Unfavorable case 2: probability of heart picked in Second draw (I.e. first draw is not Heart)= (39/52)*(13/52) =(3/4)*(1/4)= 3/16
Total Unfavorable Probability = (1/4)+(3/16) = 7/16
I.e. Favorable Probability = 1 - (7/16) = 9/16
Correct Answer: B

Approach Solution 2:
Out of 52 cards, 13 are hearts so the probability of NOT choosing a heart is
52-13 ie 39 (Favorable outcomes) divided by all possible outcomes ie 52. 39/52 = 3/4
Same for the second draw: 3/4
Total Probability = 9/16

Correct Answer: B

Approach Solution 3:

The probability of pulling a heart from a complete deck of cards is 13/52 = 1/4, hence the probability of NOT pulling a heart from a full deck is 1 - 1/4 = 3/4.
The chance of pulling a heart on the THIRD try and NOT pulling a heart on the first two attempts is (9/64).
However, the question inquired about the probability of drawing the first heart on the third draw or later. This is basically asking for the probability of NOT drawing a heart on the first two pulls, because a heart on the third, fourth, fifth, sixth, and so on draw fits our criteria.
As a result, NO HEART on the first two draws = (3/4)(3/4) = 9/16.

Correct Answer: B

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