If she made the last free throw, then her probability of making the next one is 0.9. On the other hand, If she missed the last free throw, then her probability of making the next one is 0.5.
Find the long-run free-throw shooting probabilities for this player:
To find the long-run free-throw shooting probabilities for this player, we can use conditional probabilities and the Law of Total Probability.
Let's denote the probability of making the last free throw as P(M) and the probability of missing the last free throw as P(N). We are given that P(M) = 0.9 and P(N) = 0.1.
Now, let's calculate the conditional probabilities of making the next free throw given the outcome of the last one.
If she made the last free throw (P(M)), the probability of making the next one is given as 0.9.
If she missed the last free throw (P(N)), the probability of making the next one is given as 0.5.
To find the long-run free-throw shooting probabilities, we need to consider the probability of each outcome (making or missing) for the last free throw.
The Law of Total Probability states that the probability of an event A can be found by summing the probabilities of A given different disjoint events B1, B2, ..., Bn, each weighted by the probability of each event B1, B2, ..., Bn occurring.
In this case, there are two disjoint events for the last free throw: making (M) and missing (N).
Therefore, the long-run free-throw shooting probability P(Make) can be calculated as:
P(Make) = P(M) * P(Make | M) + P(N) * P(Make | N)
Plugging in the given values:
P(Make) = 0.9 * 0.9 + 0.1 * 0.5
= 0.81 + 0.05
= 0.86
Therefore, the long-run free-throw shooting probability for this player is 0.86 or 86%.