91Ó°ÊÓ

Given in the question that, Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for$96$days.

From the information, we observed that, the sample size is $96$days.

Therefore,

$X~B(96,0.25)$

Since,$np=24>5$and$nq=72>5$

Hence,

The mean:

$\mathrm{μ}=nâ‹\dots p$

$=96â‹\dots 0.25$

$=24$

Standard Deviation

$s=\sqrt{npq}=4.24$

Now random variable $Y$has a normal distribution

$Y~N(24,4.24)$

The wanted probability is given by:

$P(X<20)=P(Y<20.5)$

$=P\left(\frac{Y−24}{4.24}<\frac{20.5−24}{4.24}\right)$

$=\mathrm{ϕ}\left(\frac{20.5−24}{4.24}\right)$

$=\mathrm{ϕ}(−0.83)$

$=1−0.7967$

$=0.2033$

Given in the question that, Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for $96$days.

Variable $X$has a binomial distribution:

$X~B(96,0.25)$

The mean:

localid="1648366098955" $\mathrm{μ}=n·p=96·0.25=24$

Standard deviation:

$s=\sqrt{npq}=4.24$

Now random variable $Y$has a normal distribution:

$Y~N(24,4.24)$

The wanted probability is given by:

$P(X>16)=P(Y>15.5)$

$=P\left(\frac{Y−24}{4.24}>\frac{15.5−24}{4.24}\right)$

$=1−\mathrm{ϕ}\left(\frac{15.5−24}{4.24}\right)$

$=1−\left(\mathrm{ϕ}\right(−2\left)\right)$

$=1−(1−\mathrm{ϕ}(2\left)\right)$

$=0.9772$

Given in the question that, Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for $96$days.

Variable $X$has a binomial distribution:

$X~B(96,0.25)$

The mean:

$\mathrm{μ}=n·p=96·0.25=24$

Standard deviation:

$s=\sqrt{npq}=4.24$

Now random variable $Y$has a normal distribution:

$Y~N(24,4.24)$

The wanted probability is given by:

$P(X=24)=P(23.5<Y<24.5)$

$=P\left(\frac{23.5−24}{4.24}<\frac{Y−24}{4.24}<\frac{24.5−24}{4.24}\right)$

$=\mathrm{ϕ}\left(\frac{24.5−24}{4.24}\right)−\mathrm{ϕ}\left(\frac{23.5−24}{4.24}\right)$

$=\mathrm{ϕ}\left(0.12\right)−\mathrm{ϕ}(−0.12)$

$=\mathrm{ϕ}\left(0.12\right)−(1−\mathrm{ϕ}(0.12\left)\right)$

$=2\mathrm{ϕ}\left(0.12\right)−1$

$=2â‹\dots 0.5478−1$

$=0.0956$