Respuesta :
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:
[tex]A = a + (A \cap B)[/tex]
In which a is the probability that he wins the first race but not the second and [tex]A \cap B[/tex] is the probability that he wins both these races.
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
The probability that he wins both races is 0.5.
This means that [tex]A \cap B = 0.5[/tex]
The probability that he wins the second race is 0.6
This means that [tex]B = 0.6[/tex]
[tex]B = b + (A \cap B)[/tex]
[tex]0.6 = b + 0.5[/tex]
[tex]b = 0.1[/tex]
The probability that he wins the first race is 0.7.
This means that [tex]A = 0.6[/tex]
[tex]A = a + (A \cap B)[/tex]
[tex]0.7 = a + 0.5[/tex]
[tex]a = 0.2[/tex]
A) he wins at least one race.
This is
[tex]P = a + b + (A \cap B) = 0.2 + 0.1 + 0.5 = 0.8[/tex]
There is an 80% probability that he wins at least one race.
B) he wins exactly one race.
This is
[tex]P = a + b = 0.2 + 0.1 = 0.3[/tex]
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.
