Question 10.6.1: a. Justify the transition probabilities for the initial stat...
a. Justify the transition probabilities for the initial state “no bases occupied.”
b. Justify the transition probabilities for the initial state “second base occupied.”
a. For the first column of A, the batter either advances to one of the bases or hits a home run. So the probability that the bases remain unoccupied is p_{H}. The batter advances to first base when the batter either walks (or is hit by a pitch) or hits a single.
Since the desired outcome can be reached in two different ways, the probability of success is the sum of the two probabilities—namely, p_{W}+p_{1}. The probabilities of the batter advancing to second base or third base are, respectively, p_{2} \text { and } p_{3}. All other outcomes are impossible, because there can be at most one runner on base after one batter when the starting state has no runners on base.
b. This concerns the third column of A. The initial state is 2:k (a runner on second base, k outs). For entry (1, 3) of A, the probability of a transition “to state 0:k” is required.
Suppose that only second base is occupied and the batter does not make an out. Only a home run will empty the bases, so the (1,3) \text {-entry is } p_{H}.
Entry (2, 3): (“to state 1:k”) To leave a player only on first base, the batter must get to first base and the player on second base must reach home plate \text { successfully. }^{5}
From Table 2, the probability of reaching home plate successfully from second base is .5. Now, assume that these two events are independent, because only the actions of the batter (and Table 2) influence the outcome. In this case, the probability of both events happening at the same time is the product of these two probabilities, so the (2,3) \text {-entry is } .5 p_{1} .
Entry (3, 3): (“to state 2:k”) To leave a player only on second base, the batter must reach second base (a “double”) and the runner on second base must score. The second condition, however, is automatically satisfied because of the assumption in Table 2. So the probability of success in this case is p_{2} \text { This is the }(3,3) \text {-entry }.
Entry \text { Entry }(4,3): (“to state 3:k”) By an argument similar to that for the (3, 3)-entry,
Entry (5, 3): (“to state 12:k”) To leave players on first base and second base, the batter must get to first base and the player on second base must remain there.
However, from Table 2, if the batter hits a single, the runner on second base will at least get to third base. So, the only way for the desired outcome to occur is for the batter to get a walk or be hit by a pitch. The (5,3) \text {-entry is thus } p_{W} .
Entry (6, 3): (“to state 13:k”) This concerns the batter getting to first base and the runner on second base advancing to third base. This can happen only if the batter hits a single, with probability p_{1}, and the runner on second base stops at third base, which happens with probability .5 (by Table 2). Since both events are required, the (6, 3)-entry is the product .5 p_{1}.
Entry (7, 3): (“to state 23:k”) To leave players on second base and third base, the batter must hit a double and the runner on second base must advance only to third base. Table 2 rules this out—when the batter hits a double, the runner on second base scores. Thus the (7, 3)-entry is zero.
Entry (8, 3): The starting state has just one runner on base. The next state cannot have three runners on base, so the (8, 3)-entry is zero.
| TABLE 2 Assumptions about Advancing Runners | |
| Outcome | Batting Event |
| The batter advances to first base. A runner on first base advances to second base. A runner on second base advances to third base only if first base was also occupied. A runner on third base scores only if first base and second base were also occupied. | Walk or Hit Batsman |
| The batter advances to first base. A runner on first base advances to second base. A runner on third base scores. A runner on second base advances to third base half of the time and scores half of the time. | Single |
| The batter advances to second base. A runner on first base advances to third base. A runner on second base scores. A runner on third base scores. | Double |
| The batter advances to third base. A runner on first base scores. A runner on second base scores. A runner on third base scores. | Triple |
| The batter scores. A runner on first base scores. A runner on second base scores. A runner on third base scores. | Home Run |
| No runners advance. The number of outs increases by one. | Out |
{ }^{5} \text { The } only other way to make the player on second base “disappear” would be for the player to be tagged out, but the model does not permit outs for runners on the bases.