# Statistical Probability of a Deck of Cards

1) Consider a regular deck of cards where the cards ace, jack, queen, and king are numbered 1, 11, 12, and 13 respectively. Consider the experiment of drawing two cards randomly from this deck without replacement.

a. Describe the outcomes of this experiment. List the elements of the sample space.

b. What is the probability of obtaining a total of 6 for the two cards?

c. Let A be the event "total card value is 4 or less". Find P(A) and P(Ac).

2) A regular six faced die is rolled 100 times, leading to the following frequencies.

Number Frequency

1 20

2 16

3 18

4 10

5 14

6 22

a. What is the probability of getting the number 2?

b. What is the probability of getting a number greater than 4?

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#### Solution Preview

1. a) We'll denote the outcomes of the two trials as ordered pairs (a,b), where a represents the number on the first card and b represents the number on the second card.

The total list of possibilities is:

(1,1), (1,2), (1,3)... (1, 13)

(2,1), (2,2), ... (2,13)

...

(13,1), (13,2), ..., (13,13)

There are 13^2 = 169 elements in the sample space!

b) The only ways to get a total of six are the following:

(1,5), ...

#### Solution Summary

The statistical probability of a deck of cards are determined.

Concepts of probability statistics

1. Event A: You spend your entire Memorial Day 2006 in Acapulco.

Event B: You spend your entire Memorial Day 2006 with some of your friends.

TRUE OR FALSE: A and B are mutually exclusive events.

2.Event A: You spend your entire Memorial Day 2004 in Denver. Event B: You spend your entire Memorial Day 2004 in Vermont.

TRUE OR FALSE: A and B are mutually exclusive events.

3. You choose a card at random from an honest deck of 52. Without looking a the card, and without replacing this card, you choose another card at random.

Event A: a black card is chosen on the first draw

Event B: a red card is chosen on the second draw.

TRUE OR FALSE: Event B is independent of Event A.

5.One hundred fifty six school children were surveyed about whether or not they had a basketball and/or a bicycle.

68 kids have only a bicycle

42 kids has both a bicycle and a basketball

35 kids had a basketball but no bicycle

11 kids has neither a bicycle nor a basketball

A child is selected at random from this group. What is the probability that the child has a bicycle, given that the child has a basketball?

a. 42/77

b. 110/145

c. 68/77

d. 35/42

e. 42/110

6. One hundred fifty six school children were surveyed about whether or not they had a basketball and/or a bicycle.

68 kids have only a bicycle

42 kids has both a bicycle and a basketball

35 kids had a basketball but no bicycle

11 kids has neither a bicycle nor a basketball

A child is selected at random from this group. What is the probability that the child has a basketball, given that the child has a bicycle?

a. 35/103

b. 77/145

c. 42/68

d. 42/77

e. 42/110

Questions 7 through 9 are based on the following survey:

200 students at a university were asked if they preferred basketball or soccer and they were asked if they preferred Snickers or M&Ms candy. (The order of the questions is not important). The survey of 200 students yielded the following results:

Snickers M&Ms Total

Basketball 88 22 110

Soccer 44 46 90

Total 132 68 200

7. Choosing one of the 200 students at random:

What is the probability student that the student prefers basketball?

Write your answers in fraction form, eg. 1/2 or 56/78. You do not need to reduce the fractions to lowest form. Do not use decimal form.

8. Choosing one of the 200 students at random:

Given that the student prefers M&Ms, what is the probability that the student prefers basketball?

9. Choosing one of the 200 students at random:

What is the probability that the student prefers basketball OR prefers M&Ms?

10. You have a regular deck of 52 cards. You remove one card. Before you look at it:

What is the probability that it is a heart?

11. You have a regular deck of 52 cards. You remove one card. Before you look at it:

What is the probability that it is red or an Ace?

12. You have a regular deck of 52 cards. You remove one card. You look at the card and see that it is the 8 of Clubs. You DO NOT put the 8 of clubs back in the deck. You choose another card at random. Before you look at it,

What is the probability that it is neither red nor an Ace?

13. Compute: 13!/9! Do not use any punctuation in your answer, i.e, no comma or decimal point.

14. Eighteen athletes are trying out for four spots on their countries Olympic team. How many possible ways can four athletes be chosen?

15. At a reception, there are six different door prizes. No one can win more than one prize. How many different ways can the prizes be awarded if there are 15 people who are eligible to win?

Do not use any punctuation in your answer, ie, no commas and no decimal point.

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