C|L means, given the person chosen is a Latino Californian, the person is a registered voter who prefers life in prison without parole for a person convicted of first degree murder.
L \cap C is the event that the person chosen is a Latino California registered voter who prefers life without parole over the death penalty for a person convicted of first degree murder.
To pick one person from the study who is Japanese American AND smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes. The sample space should include everyone in the study. The probability is \(\frac{4,715}{100,450}\).
To pick one person from the study who is Japanese American given that person smokes 21-30 cigarettes per day, means that the person must fulfill both criteria and the sample space is reduced to those who smoke 21-30 cigarettes per day. The probability is \(\frac{4715}{15,273}\).
\(P(\text { at least one green })=P(G G)+P(G Y)+P(Y G)=\frac{25}{64}+\frac{15}{64}+\frac{15}{64}=\frac{55}{64}\)
\(P(G | G)=\frac{5}{8}\)
Yes, they are independent because the first card is placed back in the bag before the second card is drawn; the composition of cards in the bag remains the same from draw one to draw two.
If we assume that all walkers are alone and that none from the other two groups travel alone (which is a big assumption) we have: \(P(\text{Alone}) = 0.7318 + 0.0390 = 0.7708\).
Make the same assumptions as in (b) we have: \((0.7708)(1,000) = 771\)
You can't calculate the joint probability knowing the probability of both events occurring, which is not in the information given; the probabilities should be multiplied, not added; and probability is never greater than 100%
A home run by definition is a successful hit, so he has to have at least as many successful hits as home runs.
The coin toss is independent of the card picked first.
\(\{(G,H) (G,T) (B,H) (B,T) (R,H) (R,T)\}\)
\(P(A)=P(\text { blue }) P(\text { head })=\left(\frac{3}{10}\right)\left(\frac{1}{2}\right)=\frac{3}{20}\)
Yes, A and B are mutually exclusive because they cannot happen at the same time; you cannot pick a card that is both blue and also (red or green). \(P(A \cap B) = 0\)
No, A and C are not mutually exclusive because they can occur at the same time. In fact, C includes all of the outcomes of A; if the card chosen is blue it is also (red or blue). \(P(A \cap C) = P(A) = \frac{3}{20}\)
No, whether the money is returned is not independent of which class the money was placed in. There are several ways to justify this mathematically, but one is that the money placed in economics classes is not returned at the same overall rate; \(P(R|E) \neq P(R)\).
No, this study definitely does not support that notion; in fact, it suggests the opposite. The money placed in the economics classrooms was returned at a higher rate than the money place in all classes collectively; \(P(R|E) > P(R)\).