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Tree diagrams · 2 marks

Can you calculate the probability of getting one head and one tail?

Answer: When a fair coin is tossed twice, ‘one head and one tail’ = {HT, TH}: 2 of the 4 equally likely outcomes. P = 24 = 12 = 0.5.

Step-by-step solution

Given: A fair coin is tossed two times; S = {HH, HT, TH, TT}
To find: P(one head and one tail)

Idea: One head and one tail can happen in two orders: head first (HT) or tail first (TH). Both paths of the tree count.

First tossSecond tossOutcome1/2H1/2HHH1/2THT1/2T1/2HTH1/2TTT
  1. From the tree diagram, the sample space is S = {HH, HT, TH, TT}: 4 equally likely outcomes.½ mark
  2. Outcomes with exactly one head and one tail: HT (head first) and TH (tail first). Number of favourable outcomes = 2.½ mark
  3. P(one head and one tail) = 24 = 12 = 0.5, or 50%.1 mark
P(one head and one tail) = 2/4 = 1/2 = 0.5 (50%).

Check: P(HH) + P(one of each) + P(TT) = 14 + 12 + 14 = 1 ✓.

Answer to write in the exam

S = {HH, HT, TH, TT}; n(S) = 4

One head and one tail = {HT, TH}; favourable outcomes = 2

∴ P(one head and one tail) = 24 = 12

Common mistakes that cost marks

  • Counting only HT and getting 14. TH is also one head and one tail.
  • Using the sample space {two heads, two tails, one of each} and answering 13. These three results are not equally likely; ‘one of each’ happens in two ways.
  • Adding branch probabilities along a path. Along a path you multiply: P(HT) = 12 × 12 = 14; then add the two paths: 14 + 14 = 12.

How this can come in the exam

Assertion–Reason (1 mark)

Assertion (A): When two fair coins are tossed, the probability of getting one head and one tail is 13.
Reason (R): When two coins are tossed, the possible results are two heads, two tails, or one of each.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(D) A is false but R is true.
A is false: the equally likely outcomes are HH, HT, TH, TT, so the probability is 24 = 12. R is true as a list of results, but those three results are not equally likely.

MCQ (1 mark)

Two fair coins are tossed. The probability of getting at most one head is

  1. 14
  2. 12
  3. 34
  4. 1
Show answer

(C) 34
At most one head: {HT, TH, TT}, 3 of 4: 34.

Try one yourself

Two fair coins are tossed. Find the probability that both coins show the same face.

Show answer

Same face: {HH, TT}. P = 24 = 12.

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