Probability & Statistics Ch.2 Section 2 Question 1

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Consider flipping two independent fair coins. Let X be the number of heads that appear. Compute P(X=x)P(X=x) for all real numbers x.

Solution

The possible outcomes from flipping two coins are:

  • Heads, Heads
  • Heads, Tails
  • Tails, Heads
  • Tails, Tails

The probability of no heads is 14\frac{1}{4}, therefore P(X=0)=14P(X=0)=\frac{1}{4}

The probability of one head is 24=12\frac{2}{4}=\frac{1}{2}, therefore P(X=1)=12P(X=1)=\frac{1}{2}

The probability of two heads is 14\frac{1}{4}, therefore P(X=2)=14P(X=2)=\frac{1}{4}

The probability of any other number of heads is zero: P(X=x)=0P(X=x)=0 for x∉{0,1,2}x\notin \{0,1,2\}