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A box contains exactly 18 cards—5 King cards, 6 Queen cards and 7 Jack cards. Three cards are drawn at random from this bag. If the three cards are drawn one by one without replacement, then what is the probability that the second card drawn is a Jack?
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A box contains exactly 18 cards—5 King cards, 6 Queen cards and 7 Jack...
Problem:
A box contains exactly 18 cards—5 King cards, 6 Queen cards, and 7 Jack cards. Three cards are drawn at random from this bag. If the three cards are drawn one by one without replacement, then what is the probability that the second card drawn is a Jack? Explain in detail.
Solution:
To find the probability that the second card drawn is a Jack, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
When three cards are drawn one by one without replacement, the total number of possible outcomes can be calculated using the concept of combinations.
The first card can be any of the 18 cards in the box. After the first card is drawn, there are 17 cards remaining. The second card can be any of the remaining 17 cards, and the third card can be any of the remaining 16 cards. Therefore, the total number of possible outcomes is given by:
total number of possible outcomes = 18 * 17 * 16 = 4896
Number of favorable outcomes:
To calculate the number of favorable outcomes, we need to consider the different cases when a Jack card can be drawn as the second card.
Case 1: The first card drawn is not a Jack.
In this case, there are 11 Jack cards remaining out of the 17 remaining cards. Therefore, the number of favorable outcomes is given by:
number of favorable outcomes (case 1) = 11 * 7 * 16 = 1232
Case 2: The first card drawn is a Jack.
In this case, there are 7 Jack cards remaining out of the 17 remaining cards. Therefore, the number of favorable outcomes is given by:
number of favorable outcomes (case 2) = 7 * 17 * 16 = 1904
Total number of favorable outcomes:
To find the total number of favorable outcomes, we need to consider both cases.
total number of favorable outcomes = number of favorable outcomes (case 1) + number of favorable outcomes (case 2)
= 1232 + 1904
= 3136
Probability:
The probability that the second card drawn is a Jack can be calculated using the formula:
probability = number of favorable outcomes / total number of possible outcomes
probability = 3136 / 4896
= 0.6406 (rounded to four decimal places)
Therefore, the probability that the second card drawn is a Jack is approximately 0.6406 or 64.06%.
A box contains exactly 18 cards—5 King cards, 6 Queen cards, and 7 Jack cards. Three cards are drawn at random from this bag. If the three cards are drawn one by one without replacement, then what is the probability that the second card drawn is a Jack? Explain in detail.
Solution:
To find the probability that the second card drawn is a Jack, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
When three cards are drawn one by one without replacement, the total number of possible outcomes can be calculated using the concept of combinations.
The first card can be any of the 18 cards in the box. After the first card is drawn, there are 17 cards remaining. The second card can be any of the remaining 17 cards, and the third card can be any of the remaining 16 cards. Therefore, the total number of possible outcomes is given by:
total number of possible outcomes = 18 * 17 * 16 = 4896
Number of favorable outcomes:
To calculate the number of favorable outcomes, we need to consider the different cases when a Jack card can be drawn as the second card.
Case 1: The first card drawn is not a Jack.
In this case, there are 11 Jack cards remaining out of the 17 remaining cards. Therefore, the number of favorable outcomes is given by:
number of favorable outcomes (case 1) = 11 * 7 * 16 = 1232
Case 2: The first card drawn is a Jack.
In this case, there are 7 Jack cards remaining out of the 17 remaining cards. Therefore, the number of favorable outcomes is given by:
number of favorable outcomes (case 2) = 7 * 17 * 16 = 1904
Total number of favorable outcomes:
To find the total number of favorable outcomes, we need to consider both cases.
total number of favorable outcomes = number of favorable outcomes (case 1) + number of favorable outcomes (case 2)
= 1232 + 1904
= 3136
Probability:
The probability that the second card drawn is a Jack can be calculated using the formula:
probability = number of favorable outcomes / total number of possible outcomes
probability = 3136 / 4896
= 0.6406 (rounded to four decimal places)
Therefore, the probability that the second card drawn is a Jack is approximately 0.6406 or 64.06%.
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A box contains exactly 18 cards—5 King cards, 6 Queen cards and 7 Jack cards. Three cards are drawn at random from this bag. If the three cards are drawn one by one without replacement, then what is the probability that the second card drawn is a Jack?
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