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CAT Quant

CAT probability practice questions

Probability questions in CAT are usually short and reward careful counting rather than heavy theory. Define the sample space, count favourable outcomes, and use the complement (one minus the probability of the opposite) whenever the phrase at least one appears. Practise the standard objects, which are coins, dice, cards and coloured balls, until counting them is automatic.

Key ideas

  • Probability = favourable outcomes / total outcomes, when outcomes are equally likely.
  • For independent events, multiply probabilities. For mutually exclusive events, add them.
  • At least one = 1 minus none.
  • Without replacement, the denominator drops with each draw.
  • A standard pack has 52 cards: 4 suits of 13, with 12 face cards and 4 aces.

Common traps

  • Adding probabilities of events that can happen together.
  • Forgetting to change the denominator when drawing without replacement.
  • Counting ordered and unordered outcomes inconsistently.

Practice questions with solutions

Try each question first, then open the solution. Aim for about 90 seconds per question.

  1. Two fair coins are tossed. What is the probability of getting at least one head?

    • A 1/2
    • B 3/4
    • C 1/4
    • D 2/3
    Show answer and solution

    Answer: B. 3/4

    All outcomes: HH, HT, TH, TT. Only TT has no head, so the probability is 1 - 1/4 = 3/4.

  2. A bag has 5 red, 3 blue and 2 green balls. One ball is drawn at random. What is the probability that it is red or green?

    • A 1/2
    • B 3/5
    • C 7/10
    • D 4/5
    Show answer and solution

    Answer: C. 7/10

    Red or green = 5 + 2 = 7 balls out of 10.

  3. Two fair dice are thrown. What is the probability that the sum is 7?

    • A 1/9
    • B 1/6
    • C 5/36
    • D 7/36
    Show answer and solution

    Answer: B. 1/6

    There are 6 ways to make 7 out of 36 outcomes, so 6/36 = 1/6.

  4. Two cards are drawn without replacement from a standard pack. What is the probability that both are aces?

    • A 1/169
    • B 1/221
    • C 1/52
    • D 4/663
    Show answer and solution

    Answer: B. 1/221

    (4/52) x (3/51) = 12/2,652 = 1/221.

  5. A fair coin is tossed 3 times. What is the probability of exactly 2 heads?

    • A 1/4
    • B 1/2
    • C 3/8
    • D 1/8
    Show answer and solution

    Answer: C. 3/8

    There are C(3, 2) = 3 favourable outcomes out of 8.

  6. A hits a target with probability 0.6 and B with probability 0.5, independently. What is the probability that the target is hit by at least one of them?

    • A 0.3
    • B 1.1
    • C 0.8
    • D 0.9
    Show answer and solution

    Answer: C. 0.8

    P(neither hits) = 0.4 x 0.5 = 0.2. So P(at least one) = 0.8.

  7. A number is chosen at random from 1 to 50. The probability that it is a multiple of 7 is p/50. What is p?

    Type in the answer (no options).

    Show answer and solution

    Answer: 7

    The multiples of 7 up to 50 are 7, 14, 21, 28, 35, 42 and 49, so p = 7.

  8. In how many distinct ways can the letters of the word MOON be arranged?

    Type in the answer (no options).

    Show answer and solution

    Answer: 12

    4! / 2! = 12, because the letter O repeats twice.

Keep practising