a survey indicates that 41% of women in the us consider reading as their favorite leisure time activity. you randomly select 4 women and ask them if reading is their favorite leisure time activity. find the probability that exactly 2 of them said yes
The easiest way to do this problem is to use a binomial probability table. If you use the table, n = 4, x = 2, and p = .41
100-41=59
41% Yes
59% No
4 C 2 = 6 (possible combos)
6(.41)^2*(.59)^4=.122216
To find the probability that exactly 2 out of 4 randomly selected women said yes to reading, we can use the binomial probability formula.
The binomial probability formula is: P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) is the probability of getting exactly x successes,
n is the total number of trials,
x is the number of successes we want to find the probability for,
p is the probability of success on a single trial,
q is the probability of failure on a single trial, which is equal to 1 - p,
and nCx (n choose x) is the number of combinations of n items taken x at a time.
In this case:
n = 4 (total number of randomly selected women)
x = 2 (number of women who said yes)
p = 0.41 (probability of a woman saying yes)
q = 1 - p = 1 - 0.41 = 0.59 (probability of a woman saying no)
Let's use the formula to calculate the probability:
To find the probability that exactly 2 out of 4 randomly selected women said yes, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
where:
P(x) is the probability of getting exactly x successes
n is the total number of trials or observations
x is the number of successes we are interested in
p is the probability of success in a single trial
In this case, n = 4 (since we are randomly selecting 4 women), x = 2 (since we are interested in exactly 2 women saying yes), and p = 0.41 (since the survey indicates that 41% of women consider reading as their favorite leisure time activity).
Now, we can plug in these values into the formula:
P(2) = (4C2) * (0.41^2) * (0.59^(4-2))
To calculate (4C2), we use the combination formula:
(4C2) = 4! / ((4-2)! * 2!) = (4 * 3) / (2 * 1) = 6
Now we can substitute everything into the formula:
P(2) = 6 * (0.41^2) * (0.59^2)
P(2) = 0.2733
Therefore, the probability that exactly 2 out of 4 randomly selected women said yes is approximately 0.2733 or 27.33%.