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%.