Chapter 9.1, Exercise 2 (Page 338)

Exercise 2:

  1. Let \(S_{200}\) be the number of heads that turn up in 200 tosses of a fair coin.

    Estimate:

    1. \(P(S_{200} =100)\)
    2. \(P(S_{200} = 90)\)
    3. \(P(S_{200} = 80)\)

Answer:

Since there are only two possible outcomes (heads and tails) this is a Bernoulli trial. Therefore, the following formula applies:

\[ \binom{n}{k}p^k(1-p)^{n-k} \]

In R we can use the dbinom() function for this, with x = the # of heads, size = the # of tosses; and prob = the probability of each outcome

# Use the dbinom function to calculate the probability
a <- dbinom(x = 100, size = 200, prob = .5)
b <- dbinom(x = 90, size = 200, prob = .5)
c <- dbinom(x = 80, size = 200, prob = .5)

cat("\nWith fair coin and 200 tosses:\n",
    "\nAnswer A: Probability of 100 =",round(a*100,1),"%",
    "\nAnswer B: Probability of 90 =",round(b*100,1),"%",
    "\nAnswer C: Probability of 80 =",round(c*100,1),"%")
## 
## With fair coin and 200 tosses:
##  
## Answer A: Probability of 100 = 5.6 % 
## Answer B: Probability of 90 = 2.1 % 
## Answer C: Probability of 80 = 0.1 %

For reference here is the probability density distribution scaled to show the probability of getting between 80 and 120 heads:

# Create sequence from 80 to 120 for X axis
x <- seq(80,120,by = 2)

# Create the binomial distribution using the above range
y <- dbinom(x,200,0.5)

# Plot the distribution
plot(x,y)