Question

Suppose you were interested in how good someone was at surfing. You find that, 54% of the

time, they successfully do a particular trick. 8% of the time, they do the trick with small

mistakes. The rest of the time they fail to do the trick at all.

- In any one attempt to do the trick, what is the probability that this person will complete

it, either perfectly or with small mistakes?

- Assume that surfing success rate in each attempt to do the trick is independent of

success in any other attempt to do the trick. Suppose you were interested in how many

times this surfer would successfully complete the trick (with no mistakes or with minor

mistakes) in four upcoming attempts. What is the probability that the surfer fails to

complete the trick in the first attempt and then completes it with no mistakes in the

next three attempts?

- What is the probability that the surfer fails to complete the trick in all four attempts

(they do not do the trick at all in any of the events)?

-What is the probability that the surfer successfully completes the trick (with or without

mistakes) in at least one of the four attempts?

-Suppose the proportion of trick success rate (completing without mistakes or with small

mistakes) was distributed on a normal curve with a mean of 0.68 and a standard

deviation of 0.07. What percent of surfers would you expect to have fewer successful

attempts (no mistakes or with small mistakes) than the surfer you watched?

Answer 1

1) Let p = P(surfer completes the trick successfully or with small mistakes)        
p = 0.54 + 0.08        
p = 0.62        
P(person will complete the trick, either perfectly or with small mistakes in any one attempt) = 0.62        
         
2) s = 0.54 Probability of success without any mistakes       
p = 0.62 Probability of success with or without mistakes       
q = 1 - 0.62 = 0.38  Probability of failure in a single attempt      
P(surfer fails to complete the trick in the first attempt and then completes it with no mistakes in the next three attempts)        
= q * s * s * s        
= 0.38 * 0.54 * 0.54 * 0.54        
= 0.0598        
P(surfer fails to complete the trick in the first attempt and then completes it with no mistakes in the next three attempts) = 0.0598        
         
3) P(surfer fails to complete the trick in all 4 attempts)        
= q * q * q * q        
= 0.38 * 0.38 * 0.38 * 0.38        
= 0.0209        
P(surfer fails to complete the trick in all 4 attempts) = 0.0209       
         
4) P(surfer successfully completes in atleast 1 of the 4 attempts)        
= 1 - P(surfer does not successfully complete in all of the 4 attempts)        
= 1 - 0.0209  (0.0209 is the probability calculated in (3))      
   = 0.9791        
P(surfer successfully completes in atleast 1 of the 4 attempts) = 0.9791        
         
5) p = 0.62   which is the probability of success of our surfer        
Let X be the proportion of trick success rate         
X follows normal distribution with mean = 0.68 and standard deviation = 0.07        
We have to find percentage of surfers with success rate less than 0.62        
P(X < 0.62)        
We use Excel function NORM.DIST to find the probability        
P(X < 0.62) = NORM.DIST(0.62, 0.68, 0.07, TRUE)        
P(X < 0.62) = 0.1957        
Percentage of surfers with fewer successful attempts than our surfer = 19.57%