Experiment: Toss a fair coin two times.
Step-by-step solution
Idea: A tree diagram lists all outcomes of a multi-step experiment: draw branches for the first toss, then from each of those draw branches for the second toss. Each path from start to end is one complete outcome.
- First toss: from a single starting point, draw a line to each possible outcome of the first toss, H and T. Each branch has probability 12.½ mark
- Second toss: from each of these outcomes draw two lines, to H and to T, again each with probability 12.½ mark
- Follow each path from start to end: H then H, H then T, T then H, T then T. Yes, the tree diagram shows 4 possible outcomes. Hence the sample space S = {HH, HT, TH, TT}.1 mark
- The 4 outcomes are equally likely. The event ‘getting Heads twice’ is the single outcome HH, so
Probability of HH = Number of favourable outcomesNumber of possible outcomes = 14 = 0.25 or 25%.1 mark
Check: Number of outcomes = 2 (first toss) × 2 (second toss) = 4 ✓. Also 12 × 12 = 14 along the HH path ✓.
Answer to write in the exam
First toss: H, T (12 each); second toss from each: H, T (12 each)
Paths: HH, HT, TH, TT, so 4 possible outcomes
S = {HH, HT, TH, TT}
∴ P(HH) = 14 = 0.25 or 25%
Common mistakes that cost marks
- Writing the sample space as {HH, HT, TT} (3 outcomes), treating HT and TH as the same. Getting H first then T is a different path from T first then H.
- Writing P(HH) = 12 + 12 = 1. Along a path the probabilities multiply (12 × 12), they do not add.
- Drawing branches for the second toss from only one of the first-toss outcomes. Every first-toss outcome must branch again.
How this can come in the exam
A fair coin is tossed twice. The probability of getting two tails is
- 14
- 13
- 12
- 34
Show answer
(A) 14
S = {HH, HT, TH, TT}; only TT is favourable: 14.
A spinner with two equal sectors, Red and Blue, is spun twice. Draw a tree diagram, write the sample space, and find the probability of getting Blue both times.
Show answer
First spin: R, B (12 each); from each, second spin R, B (12 each). S = {RR, RB, BR, BB}, 4 equally likely outcomes. P(BB) = 14.Try one yourself
A fair coin is tossed twice. What is the probability that the second toss is heads?
Show answer
Favourable: HH, TH. P = 24 = 12.
More questions like this
- Can you calculate the probability of getting one head and one tail?
- There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket.
- Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same.
- Fill in the blanks.
- In a survey of 50 students, 15 students said they liked football. The number of students who like football is 15, and the ________ (frequency/relative frequency) is __________ (fill in the fraction or decimal).