How to find the answer and provide work to help?

Question

Suppose we pick one card at random from a standard deck of 52 cards. Let K represent the event“the card is a King”; let Q represent the event“the card is a Queen”; let H represent the event“the card is a Heart”; let C represent the event“the card is a Club.”

a. Set up a two-way table to represent the 52 cards in the deck. Create three rows that categorize the cards by clubs, hearts, and other suits. Similarly, create three columns that categorize the cards by kings, queens, and other denominations.

b. Find P(K)

c. Find P(H)

d. Are the events H and C disjoint? Find (H and C)

e. Are the events K and H disjoint? Find P (K and H)

f. Find P(H or C)

g. Find P(K or H)

h. Find P(K or Q)

Answer 1

Starting with the table, you need to have 3 rows and 3 columns:

K Q #

C

H

#

After setting up the columns and rows, we need to fill in the cells. To do this, you need to consider how many kings are clubs, how many queens are clubs, and how many cards are left when we remove the king and queen of clubs. Do the same thing for hearts. For the last row, you will count up the remaining number of kings, queens, and other cards.

K Q #

C 1 1 11

H 1 1 11

# 2 2 22

We know that we have the correct number of cards because if you sum all of the numbers you will get 52.

Now, you need to go about calculating some probabilities.

To get P(K), add up the king column and divide by the total number of cards (52)

To get P(H), add up the H row and divide by the total number of cards.

For d and e, two events are disjoint when they can never happen at the same time. For d, consider whether or not it is possible to draw a single card that is both a club and a heart. For e, consider whether or not it is possible to draw a card that is both a king and a heart. If it is not possible, then the probability of both events is 0. If it is possible for them to happen at the same time, then use your chart to find where the column and the row intersect. K and H intersect in the table. Take the number in the intersection and divide by the total number of cards.

For the last three problems, use the following formula P(x or y) = P(x) + P(y) - P(x and y). Add up the x's in the table, add up the y's in the table, subtract the intersection of x and y, and divide it all by the number of cards. If x and y are disjoint, then they will not intersect, and so P(x and y) will be 0.

If you have trouble, feel free to ask for more info.