Whats the probability?

Question

The average number of days in a year where there is more than one major accident on I-10 is 48. What is the probability that there are 60 days in a year that there is more than one serious accident? 

Angelica makes 53% of her penalty kicks in soccer games. What is the probability that the first penalty kick she makes is her 4th attempt? 

A certain experimental treatment for bones work 43% of the time. What is the probability that the first time the treatment works is on the third patient? 

Answer 1

Going to use normal approximation and z-statistic to solve problem.

z = (x - mean) / standard deviation)

n = 365, p = 48/365

mean = np = 48

SD = sqrt(n*p*(1-p)) = sqrt(365*48/365*317/365) = 6.46

P(x > 60) = P(z > (60 - 48)/6.46) = P(z > 1.86) = 1 - P(z < 1.86) = 1 - 0.9686 = 0.0314

------------

The probability that she fails to make a kick is 1 - 0.53 = 0.47. She will fail the 1st three times, then succeed on the fourth:

probability = 0.47*0.47*0.47*0.53 = 0.055

-----------

The probability that the treatment fails is 1 - 0.43 = 0.57. Treatment will fail the 1st two times, then succeed on the third:

probability = 0.57 * 0.57 * 0.43 = 0.14