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Chapter 7: Problem 1

Find the probability of the given event. The coin lands heads all five times.

Short Answer

Expert verified
The probability of the coin landing heads all five times is \(1/32\) or approximately 0.03125.

Step by step solution

01

Identify the probability of a single toss

First, we need to determine the probability of the coin landing heads in a single toss. For a fair coin, there are two possible outcomes: heads and tails. Both outcomes have an equal chance of happening, so the probability of getting heads is 1/2 (one out of two possible outcomes).
02

Use the multiplication rule for independent events

Since the coin tosses are independent events, we can use the multiplication rule to find the probability of getting heads all five times. This rule states that the probability of multiple independent events occurring is the product of their individual probabilities. In this case: P(getting heads all 5 times) = P(getting heads in the 1st toss) 脳 P(getting heads in the 2nd toss) 脳 P(getting heads in the 3rd toss) 脳 P(getting heads in the 4th toss) 脳 P(getting heads in the 5th toss)
03

Calculate the probability

Using the probability of getting heads in a single toss (1/2) and applying the multiplication rule, we can now calculate the probability of getting heads all five times: P(getting heads all 5 times) = (1/2) 脳 (1/2) 脳 (1/2) 脳 (1/2) 脳 (1/2) = (1/2)^5 = 1/32 So the probability of the coin landing heads all five times is 1/32 or approximately 0.03125.

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Most popular questions from this chapter

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006, the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), B (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$\begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array}$$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are events of an experiment, then $$ P(A \cap B)=P(A \mid B) \cdot P(B)=P(B \mid A) \cdot P(A) $$

In "The Numbers Game," a state lottery, four numbers are drawn with replacement from an urn containing balls numbered \(0-9\), inclusive. Find the probability that a ticket holder has the indicated winning ticket. One digit (the first, second, third, or fourth digit)

Researchers weighed 1976 3-yr-olds from low-income families in 20 U.S. cities. Each child is classified by race (white, black, or Hispanic) and by weight (normal weight, overweight, or obese). The results are tabulated as follows: $$\begin{array}{lcccc} \hline & & {\text { Weight, \% }} & \\ \text { Race } & \text { Children } & \text { Normal Weight } & \text { Overweight } & \text { Obese } \\ \hline \text { White } & 406 & 68 & 18 & 14 \\ \hline \text { Black } & 1081 & 68 & 15 & 17 \\ \hline \text { Hispanic } & 489 & 56 & 20 & 24 \\ \hline \end{array}$$ If a participant in the research is selected at random and is found to be obese, what is the probability that the 3 -yr-old is white? Hispanic?

In a recent senatorial election, \(50 \%\) of the voters in a certain district were registered as Democrats, \(35 \%\) were registered as Republicans, and \(15 \%\) were registered as Independents. The incumbent Democratic senator was reelected over her Republican and Independent opponents. Exit polls indicated that she gained \(75 \%\) of the Democratic vote, \(25 \%\) of the Republican vote, and \(30 \%\) of the Independent vote. Assuming that the exit poll is accurate, what is the probability that a vote for the incumbent was cast by a registered Republican?
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