Quantitative Techniques · Micro-test

Probability

Probability at this level is counting: the number of outcomes you want over the number that can happen, with all outcomes equally likely.

10 questions · 5 minutes · instant scoring

What this topic actually tests

The probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes. It lies between 0 and 1, and the probability that the event does not happen is 1 minus the probability that it does. That complement is the quick route for 'at least one' questions: the chance of at least one head in two tosses is 1 minus the chance of no heads, 1 − 1/4 = 3/4. Know the standard sample spaces. One die has 6 outcomes; two dice have 36, and it is safest to list the favourable pairs. A pack of cards has 52 cards in four suits of 13, with two red suits and two black; each rank (ace, king and so on) appears four times. For 'either A or B', add the two probabilities and subtract the overlap, so that outcomes belonging to both are not counted twice: among the numbers 1 to 20 there are 6 multiples of 3 and 4 multiples of 5, but 15 is both, so 9 numbers qualify. For 'A and then B', multiply; if the first item is not put back, the second probability is worked out on what remains. Two red balls from 4 red and 6 blue, without replacement, is 4/10 × 3/9 = 2/15. Always reduce the final fraction, since the options will be in lowest terms.

The common trap on this topic

With two dice, the trap is treating the sums 2 to 12 as equally likely. They are not: a sum of 7 can be made in six ways and a sum of 2 in only one. With 'or' questions, the trap is double counting outcomes that satisfy both conditions. With draws without replacement, it is forgetting to reduce both the favourable count and the total for the second draw. And in 'at least one' questions, candidates count the favourable cases directly and miss one, where the complement would have given the answer in a line.

Probability: answers and explanations

A fair die is thrown once. What is the probability of getting a prime number?
Answer: D. 1/2
The primes on a die are 2, 3 and 5: three outcomes out of six, which is 1/2. One is not a prime number.
Two fair coins are tossed together. What is the probability of getting at least one head?
Answer: C. 3/4
The four equally likely outcomes are HH, HT, TH and TT. Only TT has no head, so the probability of at least one head is 1 − 1/4 = 3/4.
A bag contains 5 red, 3 blue and 2 green balls. One ball is drawn at random. What is the probability that it is not blue?
Answer: D. 7/10
There are 10 balls, of which 3 are blue and 7 are not. The probability is 7/10.
One card is drawn at random from a well-shuffled pack of 52 cards. What is the probability that it is a king or a queen?
Answer: B. 2/13
There are 4 kings and 4 queens, and no card is both. That is 8 cards out of 52, which reduces to 2/13.
Two fair dice are thrown together. What is the probability that the sum of the two numbers is 7?
Answer: A. 1/6
The pairs that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1): six of the 36 outcomes, which is 1/6.
Two fair dice are thrown together. What is the probability that the sum of the two numbers is 9?
Answer: A. 1/9
The pairs that sum to 9 are (3, 6), (4, 5), (5, 4) and (6, 3): four of the 36 outcomes, which is 1/9.
A bag contains 4 red and 6 blue balls. Two balls are drawn one after the other, without replacing the first. What is the probability that both are red?
Answer: B. 2/15
The first ball is red with probability 4/10. That leaves 3 red among 9 balls, so the second is red with probability 3/9. The product is 12/90 = 2/15. Four twenty-fifths would be the answer if the first ball were put back.
What is the probability that a leap year chosen at random contains 53 Sundays?
Answer: A. 2/7
A leap year has 366 days: 52 complete weeks and 2 extra days. Those two days are a consecutive pair, and there are seven possible pairs, one starting on each day of the week. Two of the seven include a Sunday: the pair starting on Saturday and the pair starting on Sunday. The probability is 2/7.
A letter is chosen at random from the letters of the word JUSTICE. What is the probability that it is a vowel?
Answer: B. 3/7
JUSTICE has seven different letters, of which U, I and E are vowels. The probability is 3/7.
A number is chosen at random from the whole numbers 1 to 20. What is the probability that it is a multiple of 3 or of 5?
Answer: C. 9/20
Multiples of 3: 3, 6, 9, 12, 15, 18 (six). Multiples of 5: 5, 10, 15, 20 (four). Fifteen is in both lists, so there are 6 + 4 − 1 = 9 such numbers, and the probability is 9/20.

FAQ

Is probability part of the CLAT syllabus?

The Consortium describes Quantitative Techniques as Class 10 mathematics and names ratios and proportions, basic algebra, mensuration and statistical estimation. Probability is not on that list, so for CLAT it is a lower priority than those topics. MH-CET Law's five-year paper has a separate Basic Mathematics section.

How many outcomes are there when two dice are thrown?

36, since each of the 6 faces of the first die can pair with each of the 6 faces of the second. The pairs (2, 5) and (5, 2) are different outcomes.

When do I add probabilities and when do I multiply?

Add for 'either A or B', subtracting any overlap. Multiply for 'A and then B', adjusting the second probability if the first outcome changes what is left.

Keep practising

Ready to see the full picture?

This micro-test covers one topic. The full CLAT diagnostic covers all five sections in 30 minutes and gives you a section-wise breakdown of where you actually stand.

Take the Full Diagnostic →