You read 8 plays over 2 days, 4 plays per day.
If Hamlet and Macbeth are two plays, what is the probability that you read them on the same day?
Hello, thefirsthokage!
Here's one approach . . .
You read 8 plays over 2 days, 4 plays per day.
If Hamlet and Macbeth are two of the plays,
what is the probability that you read them on the same day?
There are: $\displaystyle \binom{8}{4,4} = 70$ ways to read the books.
There are two cases to consider:
If you read Hamlet and Macbeth on the first day,
. . there are $\displaystyle \binom{6}{2} = 15$ choices for the other 2 books read the first day.
. . The other 4 books are read the second day.
If you read Hamlet and Macbeth on the second day,
. . there are $\displaystyle \binom{6}{2} = 15$ choices for the other 2 books read the second day.
. . The other 4 books are read the first day.
Hence, there are $\displaystyle 15 + 15 \:=\:30$ ways that Hamlet and Macbeth can be read on the same day.
Therefore: $\displaystyle P(\text{Hamlet \& Macbeth, same day}) \:=\:\frac{30}{70} \:=\:\frac{3}{7}$
Prob H read on day1 is 4/8, prob M read on day 1 given that H read on day 1
is 3/7. Therefore prob both read on day 1 is (4/8)(3/7)~=0.214.
(that is there are 4 slots out of 8 which are on day 1 to read H, but once
H is allocated to a slot on day 1 there remain 7 slots of which 3 are on
day 1 available for M.)
The prob that they are both read on dy 2 is the same, so the prob that they
are read on the same day is ~=2x0.214=0.428
RonL