Section 49 Binomial Distribution: Example

49.1 Example 1

If \(X\) denotes the number of heads from 10 tosses of a coin, where the probability of heads in any one toss is 0.7, then we write: \(X \sim Bin(10,0.7)\)

Find the probabilities for the Binomial distribution.

  • What is the probability of 3 or fewer successes i.e. \(Pr(X<=3)\)?

  • What is the probability of 4 or more successes i.e. \(Pr(X>=4)\)?

pbinom(q = 3, size=10, prob=0.7)

binom_prob <- dbinom(0:10, size=10, prob=0.7)
    
plot(0:10, binom_prob, type="h", 
     main="Bi(n=10, p=0.7)", 
     xlab="r", ylab="Prob(r)")

49.2 Example 2

Here are some further examples of binomial distributions.

binom_prob <- dbinom(0:20,size=20,prob=0.2)
plot(0:20, binom_prob, type="h", 
     main="Bi(n=20, p=0.2)", 
     xlab="r", ylab="Prob(r)")

binom_prob <- dbinom(0:20,size=20,prob=0.5)
plot(0:20, binom_prob, type="h", 
     main="Bi(n=20, p=0.5)", 
     xlab="r", ylab="Prob(r)")

binom_prob <- dbinom(0:50,size=50,prob=0.1)
plot(0:50, binom_prob, type="h", 
     main="Bi(n=50, p=0.1)", 
     xlab="r", ylab="Prob(r)")

49.3 Example 3

  • Given an unbiased coin, what is the probability of getting exactly 1 head in 5 tosses?

  • What is the probability of getting more than one head?

49.4 Example 4

  • What is the probability of getting at least one 6 from 6 throws of a dice?

49.5 Example 5

  • The proportion of people who are left-handed is accepted to be approximately 0.11.
    • In a class of 20 students how likely are there to be 0 left-handers, 1 left-hander, 2 left-handers etc?
    • What is the most likely number of left-handers?