Chapter 13.6, Problem 27E

To determine

To find: the probability of all point up from $10$ tacks drop by Mrs. Davenport.

Answer to Problem 27E

The probability of all point up is $\frac{1024}{9765625}$ .

Explanation of Solution

Given:

The probability of flipping face up is $\frac{2}{5}$ .

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 flipping face up is given by $\frac{2}{5}$ , so the probability of getting all $10$ face up is given by multiplying by $\frac{2}{5}$ for each time, that is multiply $\frac{2}{5}$ for the $10$ times.

  $\begin{array}{l}P\left(\text{facing up}\right)=\frac{4}{10}=\frac{2}{5}.\\ P\left(\text{all point up}\right)=\frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}.\\ \Rightarrow {\left(\frac{2}{5}\right)}^{10}.\end{array}$

Hence, the probability of all point up is $\frac{1024}{9765625}$ .