Chapter 13.6, Problem 4CFU

To determine

To find: the probability of getting only one $4$ if a number of cubes tossed is five times.

Answer to Problem 4CFU

The probability of getting only one $4$ is $1.2\times {10}^{-4}$ .

Explanation of Solution

Given:

The number of cubes tossed five times.

Concept used:

Binomial probability refers to the probability of exactly $x$ successes on $n$ repeated trials in an experiment which has two possible outcomes common called the binomial experiment.

If the probability of success on an individual trial is $p$ and $q$ where $p+q=1$ or $q=1-p$ .

Then the binomial probability is:

  $C{}_x{p}^x{q}^{n-x}\text{ or }C{}_x{p}^x{\left(1-p\right)}^{n-x}$ .

The probability is minimum $0$ and maximum $1$ .

  $0\le p\left(A\right)\le 1.$

If $p\left(A\right)+p\left(B\right)=1$ then $p\left(B\right)=1-p\left(A\right)$ .

At least $x$ in the probability means considering the $x$ and greater than $x$ .

At most $x$ in the probability means considering the $x$ and lower than $x$ .

Calculation:

The probability of getting a number from $1$ to $6$ is:

  $\frac{1}{6}$ .

Then the probability of getting the number $4$ from a single dice is also:

  $\frac{1}{6}$ .

If the cube is tossed $5$ times the probability of getting the only one $4$ is:

  $\begin{array}{l}\Rightarrow \frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}.\\ \Rightarrow \frac{1}{7776}.\\ \Rightarrow 1.2\times {10}^{-4}.\end{array}$

Hence, the probability of getting only one $4$ is $1.2\times {10}^{-4}$ .