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There are 13 hearts in a full deck of 52 cards. In a certain game, you

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There are 13 hearts in a full deck of 52 cards. In a certain game, you [#permalink]

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New post 28 Jun 2015, 11:27
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There are 13 hearts in a full deck of 52 cards. 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?

A 1/2
B 9/16
C 11/16
D 13/16
E 15/16
[Reveal] Spoiler: OA

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Re: There are 13 hearts in a full deck of 52 cards. In a certain game, you [#permalink]

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Hi sunita123,

Here, we're told that 13 of the 52 cards are "hearts" - this is 1/4 of the deck. We're also told that after drawing a card, the card is PUT BACK in to the deck (so there will always be 52 cards to draw from).

To draw a heart for the first time on the 3rd draw (or later), the FIRST TWO draws MUST NOT be hearts....

Probability of NOT drawing a heart on the first draw = 3/4
Probability of NOT drawing a heart on the second draw = 3/4

(3/4)(3/4) = 9/16

From this point, you will eventually draw a heart (which is what the question is asking for), so the probability is 9/16.

Final Answer:
[Reveal] Spoiler:
B


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Re: There are 13 hearts in a full deck of 52 cards. In a certain game, you [#permalink]

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New post 28 Jun 2015, 23:25
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sunita123 wrote:
There are 13 hearts in a full deck of 52 cards. 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?

A 1/2
B 9/16
C 11/16
D 13/16
E 15/16


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

Answer Option B
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Re: There are 13 hearts in a full deck of 52 cards. In a certain game, you [#permalink]

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Re: There are 13 hearts in a full deck of 52 cards. In a certain game, you   [#permalink] 15 Dec 2017, 06:19
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