Alberto figures that the probability of his hockey team winning on saturday is 65%. Alvin figures that the probability of his basketball team winning on saturday is 70%. Assume that the events are independent and that the boys estimates of the probability are accurate

A. Find the probability that both teams will win on Saturday
B. Find the probability that both teams will lose on saterday
C. Find the probability that albertos team will win but alvins team will lose
D. Find the probability that albertos team will lose but alvins team will win

Hints:

1. If the probability of the hockey team winning is 0.65, then the probability of the team losing is 1-0.65=0.35, by Kolmogorov's third axiom.
2. The multiplication rule:
The probability of two independent events happening is the product of the individual probabilities.

Example:
A. Both teams will win:
P(H∩B)=0.65*0.7=0.455

3. Since all possible outcomes are included in the four cases A,B,C, and D, their probabilities must add up to 1.0

a)45%

b)10.5%
c)19.5%
d)24.5%

To find the probability in each scenario, we need to multiply the individual probabilities together since the events are independent.

A. The probability that both teams will win on Saturday is calculated as the product of the probabilities for each team winning:
Probability(Both teams win) = Probability(Alberto's team winning) * Probability(Alvin's team winning)
= 0.65 * 0.70
= 0.455 or 45.5%

B. The probability that both teams will lose on Saturday is the complement of both teams winning. Since there are only two possible outcomes (win or lose), the probability that both teams will lose is calculated by subtracting the probability of both teams winning from 1:
Probability(Both teams lose) = 1 - Probability(Both teams win)
= 1 - 0.455
= 0.545 or 54.5%

C. The probability that Alberto's team will win but Alvin's team will lose is the product of the probability of Alberto's team winning and the probability of Alvin's team losing:
Probability(Alberto's team wins and Alvin's team loses) = Probability(Alberto's team winning) * Probability(Alvin's team losing)
= 0.65 * (1 - 0.70) (since the complement of Alvin's team winning is Alvin's team losing)
= 0.195 or 19.5%

D. The probability that Alberto's team will lose but Alvin's team will win is the product of the probability of Alberto's team losing and the probability of Alvin's team winning:
Probability(Alberto's team loses and Alvin's team wins) = Probability(Alberto's team losing) * Probability(Alvin's team winning)
= (1 - 0.65) * 0.70
= 0.455 or 45.5%