Intersection: Coins

P(A ∩ B) - both conditions satisfied

Calculator Dice Coins Marbles Cards Learn

Both Coins HeadsBoth H First H AND Second TH then T All Three SameAll Same

First Coin Heads AND Second Coin Tails

What makes this different from "one head and one tail"?

This question specifies which coin must be heads and which must be tails. We need the FIRST flip to be heads AND the SECOND flip to be tails. The outcome HT counts, but TH does not!

Why does order matter?

Think of it this way: "First heads, then tails" is a specific sequence. It's like asking for a password — "AB" and "BA" are different, even though they use the same letters.

The question as an intersection:

We need two conditions to both be true:

• Event A: First coin is heads (probability = ½)
• Event B: Second coin is tails (probability = ½)

We want P(A ∩ B) = P(A) × P(B) since the flips are independent.

All Possible Outcomes

Only HT matches (not TH - order matters!)

Step-by-Step Solution

Identify the individual probabilities

P(first coin = heads) = ½ (heads is one of two equally likely outcomes)

P(second coin = tails) = ½ (tails is one of two equally likely outcomes)

Check if events are independent

Yes! What happens on the first flip has no effect on the second flip. Each coin doesn't "know" what the other coin landed on.

Apply the multiplication rule for independent events

Result:

✓ Verification by Counting

Let's verify by counting all possibilities:

• Total outcomes: HH, HT, TH, TT = 4 outcomes
• Outcomes matching "first H AND second T": just HT = 1 outcome
• Probability = 1/4 ✓

Both methods give the same answer, confirming our result!

Key Insight: Order Changes the Probability!

"First H AND Second T" = 1/4

Only HT works

"One H AND One T (any order)" = 1/2

Both HT and TH work

By not specifying order, we allow twice as many favorable outcomes, doubling the probability!