Coins and dice: what are the chances?
✓ With solution
Three coins and two dice. A quick guess gets both questions wrong!
You toss three coins together. Each one lands heads or tails, with an equal chance of each.
What is the chance that exactly two of them land heads?
- A1 in 4
- B3 in 8
- C1 in 3
- D1 in 2
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Answer: B) 3 in 8. Three coins have 2 × 2 × 2 = 8 possible results, all equally likely. Only three of them have exactly two heads: (heads, heads, tails), (heads, tails, heads) and (tails, heads, heads). So the chance is 3 in 8. The trap is 1 in 4: it's true the number of heads can be 0, 1, 2 or 3, but those four cases aren't equally likely. Three heads can happen in only one way, while two heads can happen in three.
Now you roll two ordinary dice, each with six faces from 1 to 6.
What is the chance of getting a 6 on at least one of the dice?
- A1 in 6
- B1 in 3
- C11 in 36
- D1 in 36
Show the answer
Answer: C) 11 in 36. Two dice have 6 × 6 = 36 equally likely results. The easiest way is to count the opposite: the results with no 6 at all are 5 × 5 = 25. That leaves 36 − 25 = 11 results with at least one 6, so the chance is 11 in 36. The trap is 1 in 3, which is 1 in 6 plus 1 in 6: adding them counts the double six twice.
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