Answer:
We want to find:
Here we can use Stirling's approximation, which says that for large values of n, we get:
Because here we are taking the limit when n tends to infinity, we can use this approximation.
Then we get.
Now we can just simplify this, so we get:
And we can rewrite it as:
The important part here is the exponent, as n tends to infinite, the exponent tends to zero.
Thus:
Answer:
a) 80% probability that he wins at least one race.
b) 30% probability that he wins exactly one race.
c) 20% probability that he wins neither race.
Step-by-step explanation:
We solve this problem building the Venn's diagram of these probabilities.
I am going to say that:
A is the probability that he wins the first race.
B is the probability that he wins the second race.
C is the probability that he does not win any of these races.
We have that:
In which a is the probability that he wins the first race but not the second and is the probability that he wins both these races.
By the same logic, we have that:
The probability that he wins both races is 0.5.
This means that
The probability that he wins the second race is 0.6
This means that
The probability that he wins the first race is 0.7.
This means that
A) he wins at least one race.
This is
There is an 80% probability that he wins at least one race.
B) he wins exactly one race.
This is
There is a 30% probability that he wins exactly one race.
C) he wins neither race
Either he wins at least one race, or he wins neither. The sum of these probabilities is 100%.
From a), we have that there is an 80% probability that he wins at least one race.
So there is a 100-80 = 20% probability that he wins neither race.
Check the picture below.
