**Homework 20: Compound Probability **

**Definition **

The probability of an event is defined to be the ratio of times that you expect the event to occur after many trials:

*possible*
*outcomes*

*likely*
*equally*
*of*

*number*

*A*
*event*
*in*
*resulting*
*outcomes*

*likely*
*equally*
*of*

*number*
*A*

*P*( )

**Properties **
( )

( ) ( )

1. Imagine rolling two dice, one red and one green, many times.

a. How often do you expect to get a “4” on the red die?

b. Of the times you get a “4” on the red die, how often do you expect to get a “3” on the green die?

c. What proportion of rolls would you expect to get a “4” on the red die and a “3” on the green die?

d. What proportion of rolls would you expect to get either a “1” or a “2” on the red die?

e. What proportion of rolls would you expect to get either a “4” on the red die or a

“3” on the green die?

2. A regular deck of cards has 52 cards – 13 of each suit (clubs, diamonds, hearts, and spades) and 4 of each number (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A).

a. Find the probability of drawing an A from a complete deck.

b. Suppose you just drew an A from a complete deck, and now draw a second card.

Find the probability that the second card is at 10, J, Q, or K.

c. Find the probability of drawing an A and then a 10/J/Q/K.

d. Find the probability of drawing two cards and getting “blackjack” (an A and 10/J/Q/K, in either order).

e. Find the probability of drawing two cards and not getting “blackjack”.

3. Suppose you pull one card from a complete deck of cards.

a. What is the probability of getting either a 7 or a heart?

b. What is the probability of drawing a card that is not a 7 or a heart?

c. Draw a Venn diagram that is related to these problems.

4. Suppose you draw two cards from a deck.

a. What is the probability that both cards are hearts?

b. Find the complement of the previous probability. Describe this event in simple, everyday language.

5. Suppose you draw five cards from a deck. What is the probability that they are all the same suit?

6. Six spot Keno is a casino game with the following possible prizes (based on a $1 bet):

Prize $1 $4 $88 $1500 -$1 (lose)

Probability 12.98% 2.85% 0.31% 0.01%

a. Compute the probability of losing.

b. Freddy is going to play 6-spot Keno twice. What is the probability that he loses both times?

c. What is the probability that Freddy wins at least one prize, if he plays twice?

That is, what is the probability that he does not lose both times?

d. Suppose Freddy decides to play 10 times – what is the probability that he loses all 10 games?

e. What is the probability that Freddy wins at least one prize, if he plays 10 times?

7. Richie is betting on a horse race. There are 8 horses in the race, with equal chances of winning. Richie bet that horses A, B, and C will finish in the top three, in any order.

a. What is the probability that horse A comes in first, B comes in second, and C comes in third?

b. What is the probability that any of Richie’s three horses finishes first?

c. What is the probability that one of Richie’s horses finishes first and another of Richie’s horses finishes second?

d. What is the probability that Richie’s three horses finish in the top 3 spots?

8. After going on an Easter egg hunt, Amy’s basket has the following contents:

2 chocolate bunnies 5 chocolate eggs 1 real hard-boiled eggs 4 marshmallow bunnies

Her little brother sticks his hand into the basket and pulls out the first thing he touches.

a. What is the probability that he selects a bunny or an egg?

b. What is the probability that he selects something chocolate?

c. What is the probability that he selects some kind of egg?

d. What is the probability that he selects a chocolate egg?

e. What is the probability that he selects an egg or something chocolate?

*f. What is the probability that he does not get an egg or something chocolate? *

g. Draw a Venn diagram that relates to the previous set of problems.

h. Suppose Amy’s brother manages to reach into the basket twice and gets two treats. What is the probability that both things are chocolate?

i. What is the probability that Amy’s brother gets two treats, and neither one is chocolate? That is, the first one is not chocolate, and the second one is not chocolate.

j. What is the probability that Amy’s brother gets at least one chocolate?

9. There are six people in the school math club. Information is given about each of them:

Angela Buster Chris Deidre Edwina Fiona
6^{th} 6^{th} 6^{th} 7^{th} 7^{th} 8^{th}
Girl Boy Boy Girl Girl Girl

The club needs to choose a president. They have decided to do this by drawing a name from a hat.

a. What is the probability that a 7^{th} grader will be chosen as president?

b. What is the probability that the president is not a 7^{th} grader?

c. What is the probability that a 6^{th} grader will be chosen as president?

d. What is the probability that a 6^{th} grader or a 7^{th} grader will be chosen as
president?

e. What is the probability that a girl will be chosen as president?

f. What is the probability that the president is either a girl or a 6^{th} grader?

g. Draw a Venn diagram showing the probabilities relevant to the previous problem.

10. The math club from the previous set of problems also needs to choose a vice-president. After drawing the president’s name from the hat, they will draw a second name to be vice-

president.

a. On the back of this page, make a tree showing all of the possible president/vice- president combinations. (You may use the students’ initials, instead of writing their entire names.)

b. How many possible ways are there to fill the two offices? Show how you can answer this using multiplication.

c. How many possible outcomes have girls as both president and vice-president?

Show how you can answer this question using multiplication.

d. What is the probability that girls are chosen for both the president and the vice- president?