Question

Double fault!A professional tennis player claims to get $90\%$of her second serves in. In a recent match, the player missed $5$of her first $20$second serves. Is this a surprising result if the player’s claim is true? Assume that the player has a $0.10$probability of missing each second serve. We want to carry out a simulation to estimate the probability that she would miss $5$or more of her first $20$second serves.

a. Describe how to use a random number generator to perform one trial of the simulation. The dot plot displays the number of second serves missed by the player out of the first $20$second serves in simulated matches.

b. Explain what the dot at $6$represents.

c. Use the results of the simulation to estimate the probability that the player would miss $5$or more of her first $20$second serves in a match.

d. Is there convincing evidence that the player misses more than $10\%$of her second serves? Explain your answer.

Short answer

a. The probability of missing the second serve is $0.10$on each second serve, we expect about $1$out of $10$second serves to miss.

b. The number of serves missed in a simulated match of $20$second serves among $100$simulated matches is shown in a dot plot.

c. In the dot plot, there are six dots at $5$ and one dot at $6$, indicating that data-custom-editor="chemistry" $6$of the $100$simulated matches result in $5$or more second serve misses.

d. The fact that there are many dots in the dot plot at $2$and to the left of $2$indicates that it is highly typical to have fewer than $2$failed serves in a simulated match, implying that the player misses at most $10\%$of her second serves.

Step-by-step solution

Part (a) Step 1 : Given Information

We have to describe how to use a random generator to perform one trial of the simulation.

Part (a) Step 2 : Simplification

It is unsurprising that a professional tennis player claims to hit $90\%$of her second serves. The chance of missing a second serve is now $0.10$. As a result, we'll run a simulation to see how $20$second works. This means that

$0.10=\frac{1}{10}$

Because the chance of missing the second serve is $0.10$, we should expect around $1$out of every $10$second serves to be missed. Let's utilise a random generator to generate $20$digits at random, ranging from $0$to $90$to $9$, with zero being a miss and the other digits representing no miss.

Part (b) Step 1 : Given Information

We have to explain what the dot at $6$represents.

Part (b) Step 2 : Simplification

It is unsurprising that a professional tennis player claims to hit $90\%$of her second serves. A dot plot of the number of serves missed in a simulated match of $20$second serves among $100$simulated matches is also shown. The dot at $6$symbolises a trial in which the player failed to hit of the $20$-second serves.

Part (c) Step 1 : Given Information

We have to use the result of the simulation to estimate the probability that the player would miss five or more of her first $20$second serves in a match.

Part (c) Step 2 : Simplification

A professional tennis player claims to be able to get one of her second serves in. A dot plot of the number of serves missed in a simulated match of $20$second serves among $100$simulated matches is also provided. The dot plot then shows that there are six dots at $5$and one dot at $6$, indicating that $6$of the $100$simulated matches result in 55 or more second serve misses. Then,

$\frac{6}{100}=0.06$

We conclude that missing $5$or more of the first $20$second serves in a match has a chance of $0.06$.

Part (d) Step 1 : Given Information

We have to explain is there a convincing evidence that the player misses more than $10\%$of her second serves.

Part (d) Step 2 : Simplification

It is given that a professional tennis player claims to get $90\%$of her second serves in. And the dot plot is given which shows the number of serves missed in a simulated match of $20$second serves among $100$simulated matches. Thus, we have,

  $10\%$of the $20$second serve is $10\%\times 20=0.10\times 20$

$=2$

We then note that there are many dots in the dot plot at $2$and to the left of $2$which implies that is very common to have less than $2$missed serves in a simulated match and thus it is very likely that the player misses at most $10\%$of her second serves. Thus there is not convincing evidence that the player misses more than $10\%$of her second serves.