Question

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Coin Tossing. A balanced dime is tossed three times. The possible outcomes can be represented as follows.

Here, for example. HHT means that the first two tosses come up heads and the third tails. Find the probability that

(a) exactly two of the three tosses come up heads.

(b) the last two tosses come up tails.

(c) all three tosses come up the same.

(d) the second toss comes up heads.

Short answer

Part (a) 0.375.

Part (b) 0.250.

Part (c) 0.250.

Part (d) 0.500.

Step-by-step solution

Part (a) Step 1. Given information.

The given statement is:

A balanced dime is tossed three times. The possible outcomes can be represented as follows:

HHH, HTH, THH, TTH, HHT, HTT, THT, TTT

Part (a) Step 2. Find the probability that exactly two of the three tosses come up heads.

We know that an event's probability ranges from 0 to 1, and both 0 and 1 are included in it.

The formula for the probability of an event is:

$P\left(E\right)=\frac{No.offavorableoutcomes}{Totalno.ofoutcomes}$

The total number of outcomes of a balanced dime that is tossed three times is 8.

HHH, HTH, THH, TTH, HHT, HTT, THT, TTT

The total number of all outcomes with exactly two of the three tosses come up heads are:

HTH, THH, HHT

They are 3 in total.

Now Let's take the occurrence 'E' as exactly two of the three tosses come up heads.

The probability that exactly two of the three tosses come up heads is:

$$\begin{gathered} P\left(E\right)=\frac{3}{8} \\ =0.375 \end{gathered}$$

Part (b) Step 1. Find the probability that the last two tosses come up tails.

The total number of all outcomes with the last two tosses come up tails are:

HTT, TTT

They are 2 in total.

Now Let's take the occurrence 'E' as the last two tosses come up tails.

The probability that the last two tosses come up tails is:

$$\begin{gathered} P\left(E\right)=\frac{2}{8} \\ =0.25 \end{gathered}$$

Part (c) Step 1. Find the probability that all three tosses come up the same.

The total number of all outcomes with all three tosses comes up the same. are:

TTT, HHH

They are 2 in total.

Now Let's take the occurrence 'E' as all three tosses come up the same.

The probability that all three tosses come up the same is:

$$\begin{gathered} P\left(E\right)=\frac{2}{8} \\ =0.25 \end{gathered}$$

Part (d) Step 1. Find the probability that the second toss comes up heads.

The total number of all outcomes with the second toss comes up heads. are:

HHH, THH, HHT, THT

They are 4 in total.

Now Let's take the occurrence 'E' as the second toss comes up heads.

The probability that the second toss comes up heads is:

$$\begin{gathered} P\left(E\right)=\frac{4}{8} \\ =0.500 \end{gathered}$$