According to a national poll conducted in 2010, seventy percent of teens reported having sent or received a text message while driving. Assume that I randomly survey 14 teens and asked them whether or not they have sent or received a text message while driving.
a) What is the probability that exactly 8 out of the 14 teens had sent or received a text message while driving? Round your answer to the third decimal place.
To find the probability that exactly 8 out of the 14 teens had sent or received a text message while driving, we need to 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 having exactly x successes
- n is the total number of trials
- x is the number of successes
- p is the probability of success on a single trial
In this case, n = 14 (the number of teens surveyed), x = 8 (the number of teens who had sent or received a text message while driving), and p = 0.70 (the probability of success, i.e., the proportion of teens who reported having sent or received a text message while driving)
Now we can plug in the values and solve for P(8):
P(8) = (14C8) * (0.70)^8 * (1-0.70)^(14-8)
To calculate the binomial coefficient (14C8), we use the formula:
(14C8) = 14! / (8! * (14-8)!)
Plugging in the values, we can calculate:
(14C8) = 14! / (8! * (14-8)!) = 3003
Now, we can substitute the values into the formula:
P(8) = 3003 * (0.70)^8 * (1-0.70)^(14-8)
Calculating this expression will give us the probability that exactly 8 out of the 14 teens had sent or received a text message while driving. Rounding the answer to the third decimal place will give us the final result.