Tossing coins: one, two and three
One coin has 2 outcomes, two coins 4 (HH, HT, TH, TT) and three coins 8. List them all, then count the favourable ones.
Why HT and TH are different
Toss a ₹ 1 coin and a ₹ 2 coin together. 'Head on ₹ 1, tail on ₹ 2' is not the same as 'tail on ₹ 1, head on ₹ 2'. So HT and TH are two different outcomes. With two coins there are 4 equally likely outcomes, not 3. Tossing one coin twice gives the same 4 outcomes.
Two coins: counting events
| Event | Favourable outcomes | Probability |
|---|---|---|
| Two heads | HH | 1/4 |
| Exactly one head | HT, TH | 2/4 = 1/2 |
| At least one head | HH, HT, TH | 3/4 |
| At most one head | HT, TH, TT | 3/4 |
| No head | TT | 1/4 |
Worked example 1: two coins
Two coins are tossed together. Find the probability of getting at least one tail.
- Formula: P(E) = favourable outcomes ÷ total outcomes.
- All outcomes: HH, HT, TH, TT. Total = 4.
- 'At least one tail' means one or more tails: HT, TH, TT. That is 3 outcomes.
- P(at least one tail) = 3/4.
- Check with the complement: 'no tail' is HH only, so P = 1 − 1/4 = 3/4.
P(at least one tail) = 3/4.
Worked example 2: a coin game at a mela
At a village mela, a coin is tossed three times. Ravi wins a prize if exactly two heads appear. Find the probability that he wins, and the probability that he does not.
- Formula: P(E) = favourable outcomes ÷ total outcomes.
- All 8 outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
- Exactly two heads: HHT, HTH, THH. That is 3 outcomes.
- P(win) = 3/8.
- P(not win) = 1 − 3/8 = 5/8.
P(win) = 3/8; P(not win) = 5/8.
Key words in coin questions
- At least one head: one or more heads.
- At most one head: zero heads or one head.
- Exactly two heads: two heads, no more and no less.
- No head: all tails.
- Tossing one coin twice gives the same outcomes as tossing two coins once.
Before a cricket match, the captains toss a coin because heads and tails are equally likely. That makes the choice of batting or bowling first fair to both teams.
1 marks: “A coin is tossed twice. Find the probability of getting two tails.” Outcomes: HH, HT, TH, TT. Only TT is favourable, so P = 1/4.
2 marks: “Three coins are tossed together. Find the probability of getting at least two heads.” 8 outcomes. At least two heads: HHH, HHT, HTH, THH (4). P = 4/8 = 1/2.
2 marks: “Two coins are tossed. Find the probability of getting one head and one tail.” Favourable: HT, TH (2) of 4 outcomes. P = 2/4 = 1/2.
Coins double: 1 coin gives 2, 2 coins give 4, 3 coins give 8. Always list HT and TH separately.