A survey of adults found that 35% say that their favorite sport is professional football. You randomly select 120 adults and ask them if their favorite sport is professional football. Find the probability that more than 30 people answer yes.
To find the probability that more than 30 people answer yes, we can use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes,
n is the number of trials (in this case, the number of adults selected),
x is the number of successes (i.e., the number of people who answer yes),
p is the probability of success on a single trial (i.e., the percentage who say their favorite sport is professional football), and
(1-p) is the probability of failure on a single trial.
In this case, n = 120 (the number of adults selected), x can be any value greater than 30 (since we want to find the probability of more than 30 people answering yes), p = 35% (or 0.35), and (1-p) = 1 - 0.35 = 0.65.
Now, we need to calculate the probability of getting more than 30 people answering yes, so we need to calculate the sum of the probabilities of getting exactly 31, 32, 33, ..., 120 people answering yes.
P(more than 30) = P(31) + P(32) + P(33) + ... + P(120)
Calculating this probability manually can be quite tedious. However, we can use statistical software or calculators that have built-in functions to calculate binomial probabilities.
For instance, in Python, we can use the `scipy.stats` module to calculate binomial probabilities.
Here's an example code snippet that calculates the probability of more than 30 people answering yes using Python:
```python
from scipy.stats import binom
n = 120
p = 0.35
x = range(31, 120+1) # range of x values from 31 to 120
prob_more_than_30 = 1 - binom.cdf(30, n, p) # 1 minus the cumulative distribution function (cdf) at x = 30
for i in x:
prob_more_than_30 += binom.pmf(i, n, p)
print("The probability that more than 30 people answer yes: {:.4f}".format(prob_more_than_30))
```
By running this code, we can find the probability that more than 30 people answer yes. The output will be the probability value (rounded to four decimal places).
Remember to adjust the code based on the language or statistical software you are using.