Problem Solving
If a person runs independent trials where each trial has a 0.7 probability of success,
what is the probability that they have their first success on the 2nd trial?
This is the probability that the first trial will be unsuccessful, the second will be successful and other trials do not matter.
The probability of failure on 1st trial is 0.3, the probability of success on 2nd trial is 0.7. Hence, the probability of that the first success is on the 2nd trial is 0.3*0.7 = 0.21 (see geometric distribution for x=2)
which trial has the highest probability of being the first success?
This is the first trial, because each subsequent trial assumes that the previous trials were unsuccessful with a probability of 0.3, which will be a multiplier. Only the probability of first trial being successful is 0.7. Others are less (see geometric distribution with the highest probability, x=1).
Problem #2
In 20 independent trials, where each trial has a 0.7 probability of success,
what is the probability of getting exactly 12 successes?
This is binomial distribution and the answer is calculated in the IBM SPSS part for x=12. The answer is 0.114397
what is the most likely number of successes?
The most likely number of successes is the value of x with the highest probability. It is x=14, when p=0.191639.
Problem #3
In a 1-minute period, we expect to get 3 hits on our webpage.
What is the probability of getting 0 hits?
This is Poisson distribution. That is why the probability of getting 0 hits is 0.049787 (see Poisson, where x=0).
What is the probability of getting fewer than 3 hits?
This is either 0 hits, or 1 hit, or 2 hits. The sum of probabilities is 0.049787 + 0.149361 + 0.224042 = 0.42319
