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Equally likely outcomes · 5 marks

Which of the following experiments have equally likely outcomes? Explain.

  1. (i) A driver attempts to start a car. The car starts or does not start.
  2. (ii) Tossing a fair coin once.
  3. (iii) Rolling a fair 6-sided die.
  4. (iv) Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles.
  5. (v) A baby is born. It is a boy or a girl.
Answer: Equally likely: (ii), (iii) and (v). Not equally likely: (i) (a car usually starts) and (iv) (blue is more likely: 710 against 310).

Step-by-step solution

Idea: Outcomes are equally likely when there is no reason to expect one more often than another, for example because of symmetry (a fair coin or die) or equal numbers. If one outcome has more ways to happen, or depends on conditions, they are not equally likely.

(i) A driver attempts to start a car. The car starts or does not start.

  1. Whether a car starts depends on its condition (battery, fuel, engine). A car in normal working order starts almost every time, so “starts” is far more likely than “does not start”.½ mark
  2. Not equally likely. Having two outcomes does not make them equally likely.½ mark
Not equally likely

(ii) Tossing a fair coin once.

  1. A fair coin is symmetrical, so there is no reason for heads to come up more often than tails.½ mark
  2. Equally likely: P(H) = P(T) = 12.½ mark
Equally likely

(iii) Rolling a fair 6-sided die.

  1. A fair die is a symmetrical cube, so each of the 6 faces has the same chance of landing on top.½ mark
  2. Equally likely: each of 1, 2, 3, 4, 5, 6 has probability 16.½ mark
Equally likely

(iv) Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles.

  1. There are 10 marbles: 3 red and 7 blue. P(red) = 310, P(blue) = 710.½ mark
  2. Not equally likely: blue is more likely, because there are more blue marbles. (Each single marble is equally likely to be picked, but the colours are not.)½ mark
Not equally likely

(v) A baby is born. It is a boy or a girl.

  1. A baby is a boy or a girl with (very nearly) the same chance; there is no reason to expect one rather than the other.½ mark
  2. Equally likely: P(boy) = P(girl) = 12. (Real birth records show slightly more boys, about 51 in 100, but the two outcomes are treated as equally likely.)½ mark
Equally likely
Equally likely outcomes: (ii) tossing a fair coin, (iii) rolling a fair die, (v) a baby being a boy or a girl. Not equally likely: (i) the car starting or not (it usually starts), (iv) a red or a blue marble (P(red) = 3/10, P(blue) = 7/10).

Check: In each equally likely case, every outcome has probability 1n(S) and they add to 1: 12 + 12 = 1, 6 × 16 = 1. In (iv), 310 + 710 = 1 but the two parts differ.

Answer to write in the exam

(i)

Starting depends on the condition of the car; it usually starts.

∴ The outcomes are not equally likely.

(ii)

Fair (symmetrical) coin: S = {H, T}

∴ Equally likely; P(H) = P(T) = 12

(iii)

Fair die: S = {1, 2, 3, 4, 5, 6}

∴ Equally likely; each outcome has probability 16

(iv)

P(red) = 310, P(blue) = 710

310 ≠ 710

∴ The outcomes red and blue are not equally likely.

(v)

No reason to expect a boy more than a girl.

∴ Equally likely; P(boy) = P(girl) = 12

Common mistakes that cost marks

  • Saying (i) is equally likely because there are only two outcomes. Two outcomes are equally likely only if neither is favoured.
  • Saying (iv) is equally likely because there are two colours. There are 7 blue and only 3 red marbles.
  • Writing “yes” or “no” with no explanation. The question says “Explain”, so a reason is needed for each part.

How this can come in the exam

MCQ (1 mark)

Which of these experiments has equally likely outcomes?

  1. A student answers a question: right or wrong
  2. A spinner with 4 equal sectors is spun
  3. It rains or does not rain tomorrow
  4. A bag of 2 white and 5 black beads: a bead is picked, white or black
Show answer

(B) A spinner with 4 equal sectors is spun
Equal sectors give each sector the same chance (14). The others depend on conditions or on unequal numbers.

Assertion–Reason (1 mark)

Assertion (A): When a bead is picked at random from a bag of 4 green and 4 yellow beads, the outcomes green and yellow are equally likely.
Reason (R): The bag has the same number of green and yellow beads.

  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

(A) Both A and R are true, and R is the correct explanation of A.
P(green) = P(yellow) = 48 = 12 because the numbers are equal; R explains A.

Try one yourself

Is picking a vowel or a consonant from the letters of the word ‘IDEA’ (one letter at random) an experiment with equally likely outcomes?

Show answer

No. IDEA has 3 vowels (I, E, A) and 1 consonant (D): P(vowel) = 34, P(consonant) = 14.

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