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
217
To find the probability distribution of the number of honors obtained from selecting three cards at random without replacement, we can use a tree diagram.
First, let's establish the possible outcomes at each stage of the selection process:
1st card: There are 52 possibilities (since the pack contains 52 cards).
2nd card: There are 51 possibilities remaining (since a card has already been selected).
3rd card: There are 50 possibilities remaining (since two cards have already been selected).
Now, let's create the tree diagram:
|__H__| (1st card: Honors)
________|__A__|
| |__K__|
|
__________| (1st card: Non-Honors)
|__H__| (2nd card: Honors)
|
|__NH_| (2nd card: Non-Honors)
The probabilities for each branch can be calculated by considering the number of possibilities at each stage:
1st card - Honors (H): 4/52 (since there are 4 honors in a pack of 52 cards)
1st card - Non-Honors (NH): 48/52 (since there are 48 non-honors in a pack of 52 cards)
2nd card - Honors, after drawing Honors (HH): 3/51 (since one Honors card has already been drawn and there are 3 remaining)
2nd card - Honors, after drawing Non-Honors (NH): 4/51 (since there are 4 Honors left in the pack)
2nd card - Non-Honors, after drawing Honors (HNH): 1/51 (since one Honors card has already been drawn and there is only 1 remaining)
2nd card - Non-Honors, after drawing Non-Honors (NNH): 47/51 (since one Non-Honors card has already been drawn and there are 47 remaining)
Now, let's calculate the probabilities for the 3rd card:
3rd card - Honors, after drawing HH (HHH): 2/50 (since two Honors cards have already been drawn and there are 2 remaining)
3rd card - Honors, after drawing NH (NHH): 3/50 (since one Honors card has already been drawn and there are 3 remaining)
3rd card - Honors, after drawing HNH (HNHH): 3/50 (since one Honors card has already been drawn and there are 3 remaining)
3rd card - Honors, after drawing NNH (NNHH): 4/50 (since there are 4 Honors left in the pack)
3rd card - Non-Honors, after drawing HH (HHNH): 2/50 (since two Honors cards have already been drawn and there are 2 remaining)
3rd card - Non-Honors, after drawing NH (HNH): 2/50 (since one Honors card has already been drawn and there are 2 remaining)
3rd card - Non-Honors, after drawing HNH (NNH): 1/50 (since one Non-Honors card has already been drawn and there is only 1 remaining)
3rd card - Non-Honors, after drawing NNH (NNNH): 46/50 (since two Non-Honors cards have already been drawn and there are 46 remaining)
To find the probability distribution, we multiply the probabilities along each branch. For example, the probability of getting 0 Honors (NHNN) is (48/52) * (45/51) * (46/50) = 0.6694.
By following this process for all possible outcomes, we can obtain the complete probability distribution of the number of honors obtained.
To find the probability distribution of the number of honors obtained, we will use a tree diagram.
Step 1: Start by drawing a tree diagram with three levels, representing the three draws. Each level will have branches for each possible card that can be drawn.
Step 2: At the first level, there are 52 possibilities for the first card drawn. Label each branch with the card drawn and its probability of being drawn (1/52 for each).
Step 3: At the second level, there are 51 possibilities for the second card drawn, since one card has already been drawn without replacement. Again, label each branch with the card drawn and its probability (1/51 for each).
Step 4: At the third level, there are 50 possibilities for the third card drawn, since two cards have already been drawn without replacement. Label each branch with the card drawn and its probability (1/50 for each).
Step 5: Count the number of honors in each branch. There are four honors: A, K, Q, and J. Assign a value of 1 to each honor drawn and a value of 0 to each non-honor drawn in each branch.
Step 6: Multiply the probabilities along each branch and calculate the sum of the values of honors drawn in each branch.
Step 7: Calculate the probability distribution by dividing the sum of honors by the total probability of that branch.
Step 8: Repeat steps 5-7 for each branch in the tree diagram.
Step 9: Sum up the probabilities for each possible number of honors to obtain the probability distribution of the number of honors obtained.
By following these steps, you can use the tree diagram to find the probability distribution of the number of honors obtained from three cards drawn without replacement from a shuffled deck of 52 cards.
Cannot draw tree diagram on these posts.
Probability of picking one of those cards in each of three picks.
20/52 * (20-1)/(52-1) * (20-2)/(52-2) = ?