Three cards are drawn without replacement from a well-shuffled standard deck of 52 playing cards. Find the probability that none is a face card.
There are 12 face cards, and 40 others in a standard deck.
probability of no face cards after 3 draws is just
40/52 * 39/51 * 38/50
Three cards are selected at random without replacement, from a shuffled pack of 52 playing card. Using a tree diagram find the probability distribution of the number of honours (A,K,Q,J,10) obtained
To find the probability that none of the drawn cards is a face card, we need to determine the number of favorable outcomes and the number of possible outcomes.
Step 1: Determine the number of favorable outcomes.
In a standard deck of 52 playing cards, there are 12 face cards (4 jacks, 4 queens, and 4 kings). We need to select 3 cards, none of which are face cards. This means we need to choose 3 cards from the remaining 40 non-face cards.
The number of favorable outcomes is given by the combination formula:
C(40, 3) = 40! / (3! (40 - 3)!)
= 40! / (3! 37!)
Step 2: Determine the number of possible outcomes.
In a standard deck of 52 playing cards, there are 52 cards. We need to select 3 cards without replacement.
The number of possible outcomes is given by the combination formula:
C(52, 3) = 52! / (3! (52 - 3)!)
= 52! / (3! 49!)
Step 3: Calculate the probability.
The probability is given by the number of favorable outcomes divided by the number of possible outcomes.
P(none is a face card) = C(40, 3) / C(52, 3)
= (40! / (3! 37!)) / (52! / (3! 49!))
= (40! * (3! 49!)) / (52! * (3! 37!))
= (40 * 39 * 38) / (52 * 51 * 50)
Hence, the probability that none of the drawn cards is a face card is (40 * 39 * 38) / (52 * 51 * 50).
To find the probability that none of the drawn cards is a face card, we need to first determine the total number of possible outcomes and then the number of favorable outcomes (i.e., the outcomes where none of the cards is a face card).
Step 1: Determine the total number of possible outcomes
When three cards are drawn without replacement from a standard deck of 52 playing cards, the total number of possible outcomes can be calculated using the concept of combinations.
The number of combinations of 52 cards taken 3 at a time is given by the formula C(52, 3) which can be calculated as follows:
C(52, 3) = 52! / (3!(52-3)!)
= 52! / (3!49!)
= (52 × 51 × 50) / (3 × 2 × 1)
= 22,100.
Therefore, there are 22,100 possible outcomes when three cards are drawn without replacement from a standard deck of 52 playing cards.
Step 2: Determine the number of favorable outcomes
To find the number of favorable outcomes (i.e., the outcomes where none of the cards is a face card), we consider that there are 12 face cards in a deck (4 jacks, 4 queens, and 4 kings).
To calculate the number of ways of choosing 3 non-face cards from 40 (52 - 12), we use the combination formula.
The number of combinations of 40 cards taken 3 at a time can be calculated as follows:
C(40, 3) = 40! / (3!(40-3)!)
= 40! / (3!37!)
= (40 × 39 × 38) / (3 × 2 × 1)
= 9,880.
Therefore, there are 9,880 favorable outcomes where none of the cards is a face card when three cards are drawn without replacement from a standard deck of 52 playing cards.
Step 3: Calculate the probability
Finally, to find the probability that none of the drawn cards is a face card, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 9,880 / 22,100
= 0.446.
So, the probability that none of the drawn cards is a face card is approximately 0.446 (or 44.6%).