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Out of a pack of 52 cards one is lost; from the remainder of the pack, two cards are drawn and are found to be spades. Find the chance that the missing card is a spade.
- a)11/50
- b)11/49
- c)10/49
- d)10/50
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Out of a pack of 52 cards one is lost; from the remainder of the pack,...
Intuitively, the answer should be slightly less than 1/4.
As if you consider two cases:
1) The lost card is spades. (12 spades cards remained out of 51 cards)
2) The lost card is other suits (13 spades cards remained out of 51 cards)
The probability of having two cards drew to be spades is less for 1) than 2), as there are less spades cards for 1).
For more formal calculation:
Let H be the event that the last card is spades.
Let S be the event that the 2 cards drew are spades.
The answer to this question is: P(H|S) = P(S|H)*P(H)/P(S)
We know
P(S|H) = 12/51 * 11/50
P(H) = 1/4
P(S) = 13/52 * 12/51
Hence, the answer is ((12/51 * 11/50) * (1/4)) / (13/52 * 12/51) = 11/50
Most Upvoted Answer
Out of a pack of 52 cards one is lost; from the remainder of the pack,...
To find the chance that the missing card is a spade, we need to calculate the conditional probability.
Let's break down the problem into smaller steps:
Step 1: Find the probability of drawing two spades from the remaining cards.
Since one card is lost, there are 51 cards remaining. The probability of drawing the first spade is 13/51 (since there are 13 spades in a deck of 52 cards). After drawing the first spade, there are 12 spades left out of 50 cards. Therefore, the probability of drawing the second spade is 12/50. To find the probability of both events happening, we simply multiply the individual probabilities: (13/51) * (12/50) = 156/2550 = 6/85.
Step 2: Find the probability of the missing card being a spade.
Since one card is lost, there are 51 cards remaining. Out of these 51 cards, there are 13 spades. Therefore, the probability of the missing card being a spade is 13/51.
Step 3: Calculate the conditional probability.
The conditional probability of the missing card being a spade given that two spades were drawn is calculated using Bayes' theorem:
P(missing card is a spade | two spades drawn) = (P(two spades drawn | missing card is a spade) * P(missing card is a spade)) / P(two spades drawn)
P(two spades drawn | missing card is a spade) = (13/51) * (12/50) = 6/85 (calculated in Step 1)
P(missing card is a spade) = 13/51 (calculated in Step 2)
P(two spades drawn) = 6/85 (calculated in Step 1)
P(missing card is a spade | two spades drawn) = (6/85) * (13/51) / (6/85) = 13/51
Therefore, the chance that the missing card is a spade is 13/51.
Hence, the correct answer is option A) 11/50.
Let's break down the problem into smaller steps:
Step 1: Find the probability of drawing two spades from the remaining cards.
Since one card is lost, there are 51 cards remaining. The probability of drawing the first spade is 13/51 (since there are 13 spades in a deck of 52 cards). After drawing the first spade, there are 12 spades left out of 50 cards. Therefore, the probability of drawing the second spade is 12/50. To find the probability of both events happening, we simply multiply the individual probabilities: (13/51) * (12/50) = 156/2550 = 6/85.
Step 2: Find the probability of the missing card being a spade.
Since one card is lost, there are 51 cards remaining. Out of these 51 cards, there are 13 spades. Therefore, the probability of the missing card being a spade is 13/51.
Step 3: Calculate the conditional probability.
The conditional probability of the missing card being a spade given that two spades were drawn is calculated using Bayes' theorem:
P(missing card is a spade | two spades drawn) = (P(two spades drawn | missing card is a spade) * P(missing card is a spade)) / P(two spades drawn)
P(two spades drawn | missing card is a spade) = (13/51) * (12/50) = 6/85 (calculated in Step 1)
P(missing card is a spade) = 13/51 (calculated in Step 2)
P(two spades drawn) = 6/85 (calculated in Step 1)
P(missing card is a spade | two spades drawn) = (6/85) * (13/51) / (6/85) = 13/51
Therefore, the chance that the missing card is a spade is 13/51.
Hence, the correct answer is option A) 11/50.
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Community Answer
Out of a pack of 52 cards one is lost; from the remainder of the pack,...
Probability that the lost card is spade=13/52=1/4
probability that the lost card is not spade=39/52=3/4
probability that drawn 2 cards are spades provided lost card is spade=(12/51)*(11/50)
probability that drawn 2 cards are spades provided lost card is not spade=(13/51)(12/50)
using bayes theorem (1/4*12/51*11/50)/((1/4*12/51*11/50)+(3/4*13/51*12/50))=11/50
probability that the lost card is not spade=39/52=3/4
probability that drawn 2 cards are spades provided lost card is spade=(12/51)*(11/50)
probability that drawn 2 cards are spades provided lost card is not spade=(13/51)(12/50)
using bayes theorem (1/4*12/51*11/50)/((1/4*12/51*11/50)+(3/4*13/51*12/50))=11/50
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