Question

3 at basketball practice, julie has a 86% chance of making a shot. unfortunately, there is only a 35% chance that her coach will be watching when she shoots. a) create a tree diagram or probability area model to represent the probability of these events (makes/misses the shot & coach watches/doesnt watch). b) what is the probability that her coach will not see julie make her shot? c) is this probability a union or intersection? d) what is the probability that julie makes the shot or her coach does not watch? e) is this probability a union or intersection?

Explanation:

Step1: Define probabilities

Let $P(M)$ be the probability of making a shot, $P(M)=0.86$, so the probability of missing a shot $P(\overline{M}) = 1 - 0.86=0.14$. Let $P(W)$ be the probability that the coach is watching, $P(W)=0.35$, and the probability that the coach is not - watching $P(\overline{W})=1 - 0.35 = 0.65$.

Step2: Calculate probability for part b

The probability that the coach will not see Julie make her shot means Julie makes a shot and the coach is not watching. Using the multiplication rule for independent - events (the event of making a shot and the event of the coach watching are assumed independent), $P(M\cap\overline{W})=P(M)\times P(\overline{W})=0.86\times0.65 = 0.559$.

Step3: Determine the type for part c

The probability in part b, $P(M\cap\overline{W})$, is an intersection because we want both the event of Julie making the shot and the event of the coach not watching to occur simultaneously.

Step4: Calculate probability for part d

We use the formula for the probability of the union $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here, $A$ is the event that Julie makes the shot and $B$ is the event that the coach does not watch. So $P(M\cup\overline{W})=P(M)+P(\overline{W})-P(M\cap\overline{W})$. We know $P(M) = 0.86$, $P(\overline{W})=0.65$ and $P(M\cap\overline{W}) = 0.559$. Then $P(M\cup\overline{W})=0.86 + 0.65-0.559=0.951$.

Step5: Determine the type for part e

The probability in part d, $P(M\cup\overline{W})$, is a union because we want either Julie to make the shot or the coach not to watch (or both) to occur.

Answer:

a) A tree - diagram:

• First branch:
• Branch 1.1: Coach watches with probability $P(W)=0.35$. Then sub - branches:
• Sub - branch 1.1.1: Makes the shot with probability $P(M)=0.86$.
• Sub - branch 1.1.2: Misses the shot with probability $P(\overline{M}) = 0.14$.
• Branch 1.2: Coach does not watch with probability $P(\overline{W})=0.65$. Then sub - branches:
• Sub - branch 1.2.1: Makes the shot with probability $P(M)=0.86$.
• Sub - branch 1.2.2: Misses the shot with probability $P(\overline{M}) = 0.14$.

b) $0.559$
c) Intersection
d) $0.951$
e) Union