Problem 25 - A First Course in Probability

Question

Two coins are to be flipped. The first coin will land on heads with probability \(.6,\) the second with probability \(.7 .\) Assume that the results of the flips are independent, and let \(X\) equal the total number of heads that result. (a) Find \(P\\{X=1\\}\) (b) Determine \(E[X]\)

Step-by-Step Solution

Answer

(a) The probability of getting exactly 1 head when the two coins are flipped is P{X=1} = 0.46. (b) The expected value of the number of heads when the two coins are flipped is E[X] = 1.3.

Step 1: Probability of Coin 1 landing heads and Coin 2 landing tails

We first calculate the probability of the first coin landing on heads and the second coin landing on tails. P(Coin 1 = Head) = 0.6 P(Coin 2 = Tail) = 1 - P(Coin 2 = Head) = 1 - 0.7 = 0.3 Since the flips are independent, the probability of both events occurring is the product of their individual probabilities. P(Coin 1 = H, Coin 2 = T) = P(Coin 1 = H) * P(Coin 2 = T) = 0.6 * 0.3 = 0.18

Step 2: Probability of Coin 1 landing tails and Coin 2 landing heads

Similarly, calculate the probability of the first coin landing on tails and the second coin landing on heads. P(Coin 1 = Tail) = 1 - P(Coin 1 = Head) = 1 - 0.6 = 0.4 P(Coin 2 = Head) = 0.7 P(Coin 1 = T, Coin 2 = H) = P(Coin 1 = T) * P(Coin 2 = H) = 0.4 * 0.7 = 0.28

Step 3: Probability of Exactly 1 Head

The probability of exactly 1 head (P{X=1}) is the sum of the probabilities calculated above. P{X=1} = P(Coin 1 = H, Coin 2 = T) + P(Coin 1 = T, Coin 2 = H) = 0.18 + 0.28 = 0.46 (b) Expected Value of X (E[X])

Step 4: Probability of Each Possible Outcome

Calculate the probability for each possible number of heads that can result from flipping the coins. P{X=0}: Both coins land tails. P{X=0} = P(Coin 1 = T) * P(Coin 2 = T) = 0.4 * 0.3 = 0.12 P{X=2}: Both coins land heads. P{X=2} = P(Coin 1 = H) * P(Coin 2 = H) = 0.6 * 0.7 = 0.42

Step 5: Expected Value Calculation

Use the probabilities calculated above to find the expected value of X. E[X] = 0 * P{X=0} + 1 * P{X=1} + 2 * P{X=2} E[X] = 0 * 0.12 + 1 * 0.46 + 2 * 0.42 E[X] = 0 + 0.46 + 0.84 = 1.3