Many problems like this one can be found in the book Statistical Applications of Baseball.
The price of this short book has been reduced from $2.99 to $0.99 between June 19, 2018 to June 26, 2018. The book has some imperfection but received a favorable review from the JASA journal chance in 1997.
The book provides a useful set of statistical question for students, teachers and for people who like baseball. Hope you give it a chance!
Question: Two players have four plate appearances in every game. (I will assume all plate appearances end in either a hit or an out.)
One player has a 300 batting average (BA) against both left and right handed pitching. The other batter has a 200 batting average against left handed pitching and a 400 batting average against right handed pitching. Over a 10 game period, half the games are against right handers and half the games are against left handers.
(I am not allowing for substitution of right handers for left handers or left handers for right handers in this problem.)
What is the likelihood that the 300 hitter has at least one hit in all 10 games?
What is the likelihood that the hitter with a 200 average against left handers and a 400 average against right handers has at least one hit in all ten games?
Answer: At least one hit means 1, 2, 3, or 4 hits. It also means not 0 hits. Since 0 hits is the complement of at least one hit we can calculate the probability of one hit from:
P(at least one hit) = 1  P(zero hit),
The probability of zero hits is
P(zero hits) = (1  BA)^{4}
The chart below has the probability of zero hits for BA 200, 300, and 400.
Batting Average  P(Zero HIts)  P(at least one hit) 
0.20

0.41

0.59

0.30

0.24

0.76

0.40

0.13

0.87

The likelihood of a hitter with a batting average of 0.300 against both right and left handed pitching going 10 straight games with at least one hit is 0.76^{10 } or 0.064.
The likelihood of a hitter with a batting average of 0.200 against left handed hitters and 0.400 against right handed hitters going 10 straight games with at least one hit 0.59^{5} x 0.87^{5 }is 0.036.
Both batters are 300 hitters. However, the probability of a 10game streak is 1.77 times larger for the batter who can high both lefthanded and righthanded pitchers at 300 compared to the hitter who is 200 against left handers and 400 against right handers.
Note:
A batter that walks, gets hit by a pitch, or has a sacrifice fly or bunt appears at the plate but does not have an official at bat. In a previous post, I discuss how one calculates the probability of a getting at least on hit in a game for a batter when there are options other than hits and outs.
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