Question

you are playing a video game where your character, matt, is fighting an opponent. matt has a 25% chance that his opening attack will \crit\ (do extra damage). whether or not matt gets a crit, there are three outcomes of the first turn: win, lose, or continue fighting. the following tree diagram summarizes the probabilities:
a) the probability that matt wont get a crit on the opening attack is
b) if the opening attack crits, whats the probability that matt has to continue the fight?
c) if the attack doesnt crit, then the probability that matt doesnt win on his first turn is
d) whats the probability that matt wins on his opening turn?
e) whats the probability that matt loses on his opening turn?

Explanation:

Step1: Find probability of no - crit

The probability of getting a crit is 0.25. The probability of not getting a crit is 1 - 0.25.
$1 - 0.25=0.75$

Step2: Probability of continue given crit

From the tree - diagram, when there is a crit, the probability of continuing is 0.67.

Step3: Probability of not winning given no - crit

The probability of winning given no - crit is 0.25. So the probability of not winning given no - crit is 1 - 0.25.
$1 - 0.25 = 0.75$

Step4: Probability of winning on opening turn

We use the law of total probability. The probability of winning on opening turn is the sum of the probability of winning given a crit and the probability of winning given no - crit.
$P(\text{Win})=P(\text{Crit})\times P(\text{Win}|\text{Crit})+P(\text{No Crit})\times P(\text{Win}|\text{No Crit})$
$=0.25\times0.31 + 0.75\times0.25$
$=0.25\times(0.31 + 0.75)$
$=0.25\times1.06$
$=0.265$

Step5: Probability of losing on opening turn

We use the law of total probability. The probability of losing on opening turn is the sum of the probability of losing given a crit and the probability of losing given no - crit.
$P(\text{Lose})=P(\text{Crit})\times P(\text{Lose}|\text{Crit})+P(\text{No Crit})\times P(\text{Lose}|\text{No Crit})$
$=0.25\times0.02+0.75\times0.01$
$=0.005 + 0.0075$
$=0.0125$

Answer:

A. 0.75
B. 0.67
C. 0.75
D. 0.265
E. 0.0125