Conditional Coin Probability

P(B|A) with coin flip examples

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P(Both Heads | At Least One Head)

What is "conditional probability"?

Conditional probability is the probability of an event given that we already knowsomething else happened. The notation P(A | B) means "probability of A, given B is true."

What does "at least one head" mean?

"At least one" means one or more. So "at least one head" includes:

• Exactly one head (HT or TH)
• Exactly two heads (HH)

The only outcome NOT included is TT (zero heads).

Question

You flip two fair coins. Given that at least one coin shows Heads, what is the probability that both coins show Heads?

⚠️ This is a famous "trick question"!

Most people guess 1/2 (50%), but the correct answer is 1/3 (33.33%). Let's see why step-by-step.

Step-by-Step Solution

Define the Events

Let A = "at least one Head"

Let B = "both Heads (HH)"

We want to find P(B|A)

List All Outcomes for Two Coins

At least 1 H

No Heads

Identify the Reduced Sample Space

Given at least one Head (exclude TT):

Count Favorable Outcomes

Outcomes where both are Heads within the reduced space:

Apply the Conditional Probability Formula

Answer

Common Misconception

Many people incorrectly think the answer is 1/2. Their reasoning: "If one coin is Heads, the other coin is 50/50." But this ignores that there are three equally likely ways to get "at least one Head":

HH - Both heads (favorable)
HT - First heads, second tails
TH - First tails, second heads

Only 1 of these 3 outcomes is "both heads", so P = 1/3.