Stan the stochastic southpaw has a pitching algorithm that never changes. In the first pitch of...

Stan the stochastic southpaw has a pitching algorithm that never changes.

In the first pitch of each plate appearance, he throws a fastball 70% of the time (and therefore, an off-speed pitch 30% of the time).
The following pitch selection is determined *only* by the type of pitch immediately preceding it.
- If the previous pitch was off-speed, he will throw a fastball 75% of the time and an off-speed pitch 25% of the time.
- If the previous pitch was a fastball, he will throw either a fastball or off-speed pitch with 50% probability.

In your excitement to see Stan pitch, you spilled your soda in the concourse and missed the first two pitches of his first plate appearance of the game. You are able to see that his third pitch is a fastball. Given this information, what is the probability that the first pitch was an off-speed pitch?

Solutions

Expert Solution

Let A be the event that Stan has pitched a fastball

Let B be the event that Stan has pitched an off-speed ball

It is given that Stan has pitched a fastball on the third occasion.i.e; third pitch

There are two cases.

Case 1: The second pitch being a fastball and the first pitch an off-speed pitch

Probability of the first pitch being an off-speed pitch= 30/100

Probability of the second pitch being a fastball given that the third pitch is a fastball= 75/100

Therefore,

probability that the first pitch was an off-speed pitch given that the third pitch was a fastball=[(30/100)(75/100)]=0.225

Case 2: The second pitch being an off-speed ball and the first pitch an off-speed pitch

Probability of the first pitch being an off-speed pitch= 30/100

Probability of the second pitch being a off-speed pitch given that the third pitch is a fastball=25 /100

Therefore,

probability that the first pitch was an off-speed pitch given that the third pitch was a fastball=

[(30/100)(25/100)]=0.075

Therefore, total probability that the first pitch was an off-speed pitch

given that the third pitch was a fastball=0.225+0.075=0.3

(P.S. In the question it is given to calculate the probability of 1st pitch being an off speed pitch that is why we are calculating the probability for off-speed pitch for the 1st pitch and not the fastball, and simultaneously for the second pitch)

orchestra answered 1 year ago

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