Question

Cranky mower To start her old lawn mower, Rita has to pull a cord and hope for some luck. On any particular pull, the mower has a $20\%$chance of starting.

a. Find the probability that it takes her exactly 3 pulls to start the mower.

b. Find the probability that it takes her more than 6 pulls to start the mower.

Short answer

(a)Thus, the required probability is 0.128

(b)Thus, the required probability is 0.24

Step-by-step solution

Part (a) Step 1: Given Information

Probability of success $\left(p\right)=20\%=0.20$

Formula used:

The formula to compute the geometric distribution is:

$P(X=x)=(1-p{)}^{x-1}p$

Part (a) Step 2: Simplificaiton

Consider, X be the random variable that follows the geometric distribution.

$\begin{array}{rcl}P(X& =& 3)={(1-0.20)}^{\{3-1\}}(0.20)\\ & =& 0.128\end{array}$

Thus, the required probability is $0.128$.

Part (b) Step 1: Given information

Probability of success$\left(p\right)=20$percent

Part (b)  Step 2: Calculation

The probability that it will take more than 6 pulls can be calculated as:

$\begin{array}{rcl}P(X& >& 6)=1-P(X\le 6)\\ & =& 1-\left[P\right(X=1)+P(X=2)+\dots .+P(X=6\left)\right]\\ & =& 1-[0.22+0.17+0.13+0.10+0.08+0.06]\\ & =& 0.24\end{array}$

Thus, the required probability is $0.24$