Question

1. a bag has 4 quarters, 5 pennies, and 6 nickels. what is the probability that i reach in the bag, grab a quarter, do not replace it, and then grab a penny?
2. i surveyed 64 students, 15 of them like nike, 17 of them like adidas, 20 of them like new balance, and 11 of them like nike and new balance. what is the probability that i pick a random student who likes nike or new balance?
3. at garfield, 50% of teachers ride their bike to school, 10% of teachers swim to school, and the remaining 40% drive a car to school. what is the probability that i pick a teacher who drives or swims to school?

Explanation:

1.

Step1: Calculate probability of grabbing a quarter first

The total number of coins is $4 + 5+6=15$. The probability of grabbing a quarter first, $P(Q_1)=\frac{4}{15}$.

Step2: Calculate probability of grabbing a penny second

After grabbing a quarter and not - replacing it, there are $15 - 1 = 14$ coins left. The probability of grabbing a penny second, $P(P_2)=\frac{5}{14}$.

Step3: Calculate the combined probability

Since these are dependent events, the probability of both events occurring is $P = P(Q_1)\times P(P_2)=\frac{4}{15}\times\frac{5}{14}=\frac{20}{210}=\frac{2}{21}$.

Step1: Use the formula for $P(A\cup B)$

The formula for the probability of $A$ or $B$ is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here, let $A$ be the event that a student likes Nike and $B$ be the event that a student likes New - Balance. $P(A)=\frac{15}{64}$, $P(B)=\frac{20}{64}$, and $P(A\cap B)=\frac{11}{64}$.

Step2: Substitute values into the formula

$P(A\cup B)=\frac{15 + 20-11}{64}=\frac{24}{64}=\frac{3}{8}$.

Step1: Identify the probabilities of driving and swimming

The probability that a teacher drives, $P(D)=0.4$ and the probability that a teacher swims, $P(S)=0.1$.

Step2: Use the formula for $P(A\cup B)$ for mutually - exclusive events

Since a teacher cannot drive and swim to school at the same time (mutually - exclusive events), $P(D\cup S)=P(D)+P(S)$.

Step3: Calculate the probability

$P(D\cup S)=0.4 + 0.1=0.5$.

Answer:

$\frac{2}{21}$

2.