Maths(四)---Probability and Statistics(7)

The Probability Distribution of a Binomial Random Variable

例一.

A fair coin is tossed 8 times. Find the probability of obtaining 5 heads

Solution

Let X be the number of heads obtained in 8 tosses.Then X~Bin(8,1/2)

例二.

There are 10 multiple-choice questions in a test and each question has 5 options.Suppose a student answers all 10 questions by randomly picking an option in eachquestion.

Find the probability that(a) he will answer six questions correctly,(b) he will get at least 3 correct answers.

Solution

Let X be the number of correct answers he will get.Then X ~ Bin(10, 0.2).

例三.

Binary digits 0 and 1 are transmitted along a data channel in which the presence ofnoise results in the fact that each digit may be wrongly received with a probabilty of0.00002. Each message is transmitted in blocks of 2000 digits.

(a) What is the probabilty that at least one digit in a block is wrongly received?

(b) If a certain message has a length of 20 blocks, find the probability that 2 ormore blocks are wrongly received.

Solution

(a) Then X ~ Bin(2000, 0.00002).

(b) Then Y~ Bin(20, 0.0392).