Chapter 15

A committee of three is chosen at random from a group of four women and five men. Find the probability that the committee contains at least one man. \(\frac{20}{21}\)

How many different five-card hands can be dealt from a deck of 52 playing cards? \(2,598,960\)

Ahmed, Bob, Carl, Dan, Ed, Frank, Gino, Harry, Julio, and Mike are randomly divided into two five-man teams for a basketball game. What is the probability that Ahmed, Bob, and Carl are on the same team?

How many different seven-card hands can be dealt from a deck of 52 playing cards? \(133,784,560\)

A contractor estimates a probability of \(0.7\) of making \(\$ 20,000\) on a building project and a probability of \(0.3\) of losing \(\$ 10,000\) on the project. What is his mathematical expectation? \(\$ 11,000\)

Seven coins are tossed. Find the probability of getting four heads and three tails. \(\frac{35}{128}\)

How many different five-person committees can be formed from a group of 50 people? \(2,118,760\)

A farmer estimates his corn crop at 30,000 bushels. On the basis of past experience, he also estimates a probability of \(\frac{3}{5}\) that he will make a profit of \(\$ 0.50\) per bushel and a probability of \(\frac{1}{5}\) of losing \(\$ 0.30\) per bushel. What is his expected income from the corn crop?

Nine coins are tossed. Find the probability of getting three heads and six tails.

How many different juries consisting of 11 people can be chosen from a group of 30 people? \(\quad 54,627,300\)

Bill finds that the annual premium for insuring a stereo system for \(\$ 2500\) against theft is \(\$ 75\). If the probability that the set will be stolen during the year is \(0.02\), what is Bill's expected gain or loss by taking out the insurance? \(-525\)

Six coins are tossed. Find the probability of getting at least four heads. \(\frac{11}{32}\)

How many seven-person committees consisting of three juniors and four seniors can be formed from 45 juniors and 53 seniors? \(4,155,186,750\)

Sandra finds that the annual premium for a \(\$ 2000\) insurance policy against the theft of a painting is \(\$ 100\). If the probability that the painting will be stolen during the year is \(0.01\), what is Sandra's expected gain or loss in taking out the insurance? \(-\$ 80\)

Five coins are tossed. Find the probability of getting no more than three heads. \(\frac{13}{16}\)

If the probability of some event happening is \(0.4\), what is the probability of the event not happening? Explain your answer.

Each arrangement of the 11 letters of the word MISSISSIPPI is put on a slip of paper and placed in a hat. One slip is drawn at random from the hat. Find the probability that the slip contains an arrangement of the letters with the four S's at the beginning. \(\frac{1}{330}\)

Explain each of the following concepts to a friend who missed class the day this section was discussed: usingcomplementary events to determine probabilities, using union and intersection of sets to determine probabilities, and using expected value to determine the fairness of a game.

Each arrangement of the seven letters of the word OSMOSIS is put on a slip of paper and placed in a hat. One slip is drawn at random from the hat. Find the probability that the slip contains an arrangement of the letters with an \(\mathrm{O}\) at the beginning and an \(\mathrm{O}\) at the end. \(\frac{1}{21}\)

What are the odds in favor of getting three heads with a toss of three coins? 1 to 7

Consider all possible arrangements of three identical H's and three identical T's. Suppose that one of these arrangements is selected at random. What is the probability that the selected arrangement has the three H's in consecutive positions? \(\frac{1}{5}\)

What are the odds against getting four tails with a toss of four coins? 15 to 1

\text { Explain the concepts of sample space and event space. }

What are the odds against getting three heads and two tails with a toss of five coins? 11 to 5

Explain the concepts of sample space and event space.

How would you explain the concept of conditional probability to a classmate who missed the discussion of this section?

What are the odds in favor of getting a sum of 5 with one toss of a pair of dice? 1 to 8

How would you give a nontechnical description of conditional probability to an elementary algebra student?

What are the odds against getting a sum greater than 5 with one toss of a pair of dice? 5 to 13

Explain in your own words the concept of independent events.

Suppose that one card is drawn at random from a deck of 52 playing cards. Find the odds against drawing a red card. 1 to 1

Suppose that one card is drawn at random from a deck of 52 playing cards. Find the odds in favor of drawing an ace or a king. 2 to 11

If \(P(E)=\frac{4}{7}\) for some event \(E\), find the odds in favor of \(E\) happening. 4 to 3

If \(P(E)=\frac{5}{9}\) for some event \(E\), find the odds against \(E\) happening. 4 to 5

Suppose that there is a predicted \(40 \%\) chance of freezing rain. State the prediction in terms of the odds against getting freezing rain. 3 to 2

Suppose that there is a predicted \(20 \%\) chance of thunderstorms. State the prediction in terms of the odds in favor of getting thunderstorms. 1 to 4

If the odds against an event happening are 5 to 2 , find the probability that the event will occur.

The odds against Belly Dancer winning the fifth race are 20 to 9 . What is the probability of Belly Dancer winning the fifth race? \(\quad P=\frac{9}{29}\)

The odds in favor of the Mets winning the pennant are stated as 7 to 5 . What is the probability of the Mets winning the pennant?