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Solved: Independent trials that result in a success with
Chapter 3, Problem 16TE(choose chapter or problem)
Problem 16TE
Independent trials that result in a success with probability p and a failure with probability 1 − p are called Bernoulli trials. Let Pn denote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that
and use this formula to prove (by induction) that
Questions & Answers
QUESTION:
Problem 16TE
Independent trials that result in a success with probability p and a failure with probability 1 − p are called Bernoulli trials. Let Pn denote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that
and use this formula to prove (by induction) that
ANSWER:
Step 1 of 2
(a)
The independent trials that result in a success with probability and a failure with probability
are called Bernoulli trials.
We are asked to prove that
Let denote the probability that
Bernoulli trials result in an even number of successes.
If the first trial is a success then we have trials to go and to obtain an even total number of tosses we want the number of successes in the
trials to be odd This occurs with probability
.
If the first trial is a failure then we have trials to go and to obtain an even total number of tosses we want the number of successes in the
trials to be even This occurs with probability
.
Hence we can write
Where is the probability of success.
Hence proved.