player A or player B ?

Question

THINGS ARE NOT WHAT THEY SEEM (Calculus problem)
Player A has higher batting average that player B for the first half of the baseball season.player A also has higher batting average that player B for the 2nd half of the season.Prove or disprove that player A has a higher batting average that player B at the end of the season.

Answer 1

There is no way to tell if player A has a higher batting average or not, since BA is a ratio. For example:

1st half of season
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player A bats once and gets 1 hit: 1000 BA
player B bats 100000 times and gets 99999 hits: 999.99 BA

2nd half of season
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player A bats twice and gets 1 hit: 500 BA
player B bats 100000 times and gets 49999 hits: 499.99 BA

In both halves, player A has a higher BA. His overall BA is:

(1+1) / (1+2) = 2/3 = 667 BA

Player B has:

(99999 + 49999) / (100000 + 100000) = 149998/200000 = 749.99 BA

So player B still has the better BA even though he had a lower BA each half.

Answer 2

I don't think this is a calculus question; it's a question of weighted averages.

If player A has batting average A1 for the first half and A2 for the second half, then his average for the whole season will be between A1 and A2.

Likewise, player B's whole-season average will be between B1 and B2.

If A1>B1 and A2>B2, then the average of A1 and A2 beats the average of B1 and B2. But the simple "add and divide by 2" average is not appropriate for this problem.

I'm sorry I can't draw here, or I might be able to make it clear where this counterexample comes from:

First half of season: Player A bats .500 in 10 at-bats, while player B strikes out in his only at-bat. Player A has the advantage.

Second half of season: Player A bats 1.000 in his only attempt, while Player B bats .800 in 10 at-bats. Again, advantage goes to A.

When we look at whole-season stats, Player A got on base 6 times in 11 tries (5 in the first half, 1 in the second), while Player B got on base 8 times in 11 tries (none in the first half, 8 in the second). Overall, Player B has the higher whole-season batting average.

Answer 3

Huh!!!