Question

1 multiple choice 2 points
if a fair coin is tossed four times, the probability of tossing exactly three heads is
1/16
1/4
1/2
1/8
2 multiple choice 2 points
out of a bag containing marbles of three different colors, 40 are drawn. the frequency of different colors drawn is shown in the table. based on this table what would be the probability of drawing a red marble?
color frequency
black 20
red 15
white 5
3/4
1/4
3/8
1/2

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successful trials, $p$ is the probability of success in a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. For coin - tossing, $n = 4$, $k = 3$, and $p=\frac{1}{2}$.
First, calculate the combination $C(4,3)$:
$C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!×1}=4$

Step2: Calculate the probability

$p=\frac{1}{2}$, $1-p=\frac{1}{2}$, $n = 4$, $k = 3$.
$P(X = 3)=C(4,3)\times(\frac{1}{2})^{3}\times(1-\frac{1}{2})^{4 - 3}$
$P(X = 3)=4\times(\frac{1}{2})^{3}\times\frac{1}{2}$
$P(X = 3)=4\times\frac{1}{8}\times\frac{1}{2}=\frac{4}{16}=\frac{1}{4}$

Step3: For the second problem, use the probability formula for relative frequency

The probability of an event $E$ is given by $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. The total number of marbles drawn is $20 + 15+5=40$. The number of red marbles drawn is $15$.
$P(\text{red})=\frac{15}{40}=\frac{3}{8}$

Answer:

1. B. $\frac{1}{4}$
2. C. $\frac{3}{8}$