Laura has a Deck of Standard Playing Cards with 13 of the 52 Cards GMAT Problem Solving

Question: Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart." If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?

(A) 1/4
(B) 1/5
(C) 5/26
(D) 12/42
(E) 13/42

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

The problem statement states that:

Given:

• Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart."
• Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck.

Find Out:

• The probability that the 10th card dealt is a heart.

As per the condition of the question, desired outcomes= 13
Total outcomes= 52
Probability of getting any card as the heart card= 13/52= 1/4
It will be the same probability for the 10th card as well.

Therefore, the probability that the 10th card dealt is a heart = 1/4.

Approach Solution 2:

The problem statement informs that:

Given:

• Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart."
• Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck.

Find Out:

• The probability that the 10th card dealt is a heart.

Let us use the concept of reverse probability to solve the above mentioned problem.
The probability that the 10th card drawn is a heart will be the same as (1- the probability that the 10th card drawn is not a heart)
Let us denote the probability that the 10th card drawn is not heart as P(Not H) and the probability that the 10th card drawn is heart as P(H)

Then, P(H) = 1- P (Not H)

Now the 10th card has 39 options if it is not a heart.

Hence, P (Not H) = 39/52
Or, P (Not H) = 3/4
So, P(H) = 1 - 3/4
Or, P(H) = 1/4

Therefore, the probability that the 10th card dealt is a heart = 1/4.

Approach Solution 3:

The problem statement implies that:

Given:

• Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart."
• Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck.

Find Out:

• The probability that the 10th card dealt is a heart.

Let us use the approach of permutation and combination to solve the problem.
We can see the problem in the following way:

Our required probability = (Total no. of ways in which the 10th card is a heart) / (Total no. of ways in which the 10th card can be drawn)

The total no. of ways in which the 10th card can be drawn is (\(^{52}C_1\) ) which is equal to
52! / [1!*(52-1)!] = 52! / [1*51!] = 52
Please note that we have used the formula (\(^nC_r\) ) = n!/ [r! * (n-r)!]

Similarly, the numerator will be the total no. of ways in which the 10th card is a heart (\(^{13}C_1\) ) which is equal to 13! / [1!*(13-1)!] = 13! / [1*12!] = 13
Hence, the required probability is 13/52
The probability that the 10th card drawn is 13/52 i.e. 1/4

Therefore, the probability that the 10th card dealt is a heart = 1/4.

“Laura has a deck of standard playing cards with 13 of the 52 cards”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT Problem Solving questions help the candidates to improve their mathematical knowledge in order to crack numerical problems. GMAT Quant practice papers assist the candidates to analyse varieties of questions that will enable them to score better in the GMAT exam.

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