Chapter 6.2, Problem 68E

To determine

Probability for Barsha beats Andrea in the 400 − meter race on a randomly selected day.

Answer to Problem 68E

Probability that Barsha beats Andrea in the 400 − meter race is 0.3483.

Explanation of Solution

Given information:

For A :

Mean,

  ${\mu }_A=62\text{seconds}$

Standard deviation,

  ${\sigma }_A=0.8\text{seconds}$

For B :

Mean,

  ${\mu }_B=62.8\text{seconds}$

Standard deviation,

  ${\sigma }_B=1\text{second}$

Calculations:

A: Andrea’s time in the 400 − meter race

B: Barsha’s time in the 400 − meter race

Then

A-B is the difference in time it takes each individual to complete their 400 − meter race.

A andB are independent.

Since A and B have a Normal distribution, their difference ( A - B ) also has a Normal distribution.

Now,

For any two variables,

The mean of the difference is equal to the difference of their means.

  ${\mu }_{A-B}={\mu }_A-{\mu }_B=62-62.8=-0.8\text{seconds}$

When the random variables are independent, the variance of the difference is equal to the sum of their variances.

  ${\sigma }_{A-B}^2={\sigma }_A^2+{\sigma }_B^2={\left(0.8\right)}^2+{\left(1\right)}^2=0.64+1=1.64\text{seconds}$

We know that

The standard deviation is the square root of the variance.

  ${\sigma }_{A-B}=\sqrt{{\sigma }_{A-B}^2}=\sqrt{1.64}\approx 1.2806\text{seconds}$

Thus,

The difference ( A - B ) follows a Normal distribution with mean of -0.8 seconds and standard deviation of approx. 1.2806 seconds.

Now,

For Probability:

Since Barsha beats Andrea.

This implies

  $A>B\left(\text{or}A-B>0\right)$

Then

We need to consider $x=0$ .

Calculate the z − score,

  $z=\frac{x-\mu }{\sigma }=\frac{0-\left(-0.5\right)}{1.2806}\approx 0.39$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with 0.3 and the column that starts with .09 of the standard normal probability table for $P\left(z<0.39\right)$ .

  $\begin{array}{l}P\left(A-B>0\right)=P\left(z>0.39\right)\\ =1-P\left(z<0.39\right)\\ =1-0.6517\\ =0.3483\end{array}$

Thus,

Probability that Barsha beats Andrea in the 400 − meter race on a randomly selected day is 0.3483.