Question

Tennis When a tennis player hits his first serve as hard as possible (called a blast), he gets the ball in (that is, within bounds) 60% of the time. When the blast first serve is in, he wins the point 80% of the time. When the first serve is out, his gentler second serve wins the point 45% of the time. Draw a tree diagram representing the probabilities of winning the point for the first two serves. Use the tree diagram to determine the probability that the server eventually wins the point when his first serve is a blast.

Answer 1

To represent the probabilities of winning the point for the first two serves and determine the probability of winning the point when the first serve is a blast, we can use a tree diagram. Here's how it would look:

```

┌── Blast Serve In (0.6) ── Win (0.8)

│

Start ─ Blast Serve (0.4) ┤

│

└── Blast Serve Out (0.4) ── Second Serve (0.45) ── Win (0.45)

```

In the tree diagram:

- "Start" represents the starting point of the game.

- "Blast Serve" represents the first serve being hit as hard as possible.

- "Blast Serve In" represents the event where the first serve is in.

- "Blast Serve Out" represents the event where the first serve is out.

- "Second Serve" represents the event where the player goes for a gentler second serve after the first serve is out.

- "Win" represents winning the point.

The probabilities associated with each branch are indicated next to the corresponding branches.

To determine the probability of winning the point when the first serve is a blast, we follow the path from the Start to Blast Serve and then to Blast Serve In and finally to Win. The probability is calculated by multiplying the probabilities along the path:

Probability of winning with a blast serve = Probability of Blast Serve * Probability of Blast Serve In * Probability of Win

Probability of winning with a blast serve = 0.4 * 0.6 * 0.8 = 0.192 or 19.2%

Therefore, the probability that the server eventually wins the point when his first serve is a blast is 19.2%.

Answer 2

Let's draw a tree diagram to represent the probabilities of winning the point for the first two serves.

_______Blast (60%)
|
|_______In (80%)
|
First Serve (40%) ----|
|_______Out (20%)
|
_______Gentle (40%)
|
|_______In (45%)
|
|_______Out (55%)

In this tree diagram, we have three branches representing the three possible choices for the first serve: Blast, In, and Out. From there, we have two branches originating from each choice, representing the win or loss of the point.

Now, to determine the probability that the server eventually wins the point when his first serve is a blast, we need to follow the Blast branch in the tree diagram.

The probability of the first serve being a Blast is 60%. From there, the probability of the Blast serve being In and winning the point is 80%.

Therefore, to find the overall probability of winning the point when the first serve is a Blast, we multiply the probabilities along the path: 60% (Blast) * 80% (In) = 48%.

So, the probability that the server eventually wins the point when his first serve is a Blast is 48%.

I hope this helps.