Find the probability that the product of four numbers is divisible by 5 or 10.

Correct — option (2), 369/625.

The stem does not say how the four numbers are chosen, so no probability follows from its words alone. This card assumes what the stem leaves unstated: each number's last digit is equally likely to be any digit from 0 to 9, independently of the other three. Under that assumption the answer is 369/625.

Step 1 — merge the two conditions. Every multiple of 10 is also a multiple of 5, so "divisible by 5 or 10" is the same event as "divisible by 5".

Step 2 — use the fact that 5 is prime. A product is divisible by 5 only if at least one of its factors is divisible by 5.

Step 3 — one number. It is a multiple of 5 when its last digit is 0 or 5. Under the assumption, that is 2 equally likely digits out of 10.
P(multiple of 5) = 2/10 = 1/5
P(not a multiple of 5) = 1 − 1/5 = 4/5

Step 4 — none of the four numbers is a multiple of 5.
P(none) = 4/5 × 4/5 × 4/5 × 4/5 = 256/625

Step 5 — take the complement.
P(at least one) = 1 − 256/625 = (625 − 256)/625 = 369/625

The idea to remember: a product is divisible by a prime p exactly when at least one factor is, so the probability is 1 − P(no factor divisible by p).

Why the others are wrong

• (1)
  256/369 — 256/369 divides the probability that no number is a multiple of 5 (256/625) by the probability that at least one is (369/625). That ratio is the odds against the event, not its probability.

  A probability divides by all cases. Under the card's assumption each remainder on division by 5 is equally likely, so four numbers give 5⁴ = 625 equally likely cases, of which 369 make the product divisible by 5.

• (3)
  273/2500 — 273/2500 is about 0.11. Under the card's assumption that is smaller than 1/5, the chance that the first number alone is a multiple of 5.

  The product is divisible by 5 whenever any one of the four numbers is, so under that assumption the probability cannot be less than 1/5. The complement method gives 369/625, about 0.59.

• (4)
  256/625 — 256/625 = (4/5)⁴ is the probability that none of the four numbers is a multiple of 5, the opposite of the event asked.

  It also happens to equal the chance that exactly one number is a multiple of 5: 4 × 1/5 × (4/5)³ = 256/625. The product is divisible by 5 when at least one number is, so the answer is 1 − 256/625 = 369/625.

The complement rule: P(at least one) = 1 − P(none). It turns a count of many cases (exactly one, two, three or four) into one case.

For independent events, the probability that all of them happen is the product of their probabilities. Under this card's assumed model, (4/5)⁴ is the chance that none of four numbers is a multiple of 5.

A prime p divides a product only if it divides at least one factor. And when one event sits inside another (every multiple of 10 is a multiple of 5), their union is simply the larger event.

Key facts

• A whole number is divisible by 5 exactly when its last digit is 0 or 5.
• Every multiple of 10 is a multiple of 5, so "divisible by 5 or 10" is the same event as "divisible by 5".
• Because 5 is prime, a product is divisible by 5 only if at least one factor is.
• P(at least one) = 1 − P(none); under the assumed last-digit model, 1 − (4/5)⁴ = 1 − 256/625 = 369/625.
• Under the same model, the product of four numbers is divisible by 10 with probability 1107/2000.

Common traps

• Adding a probability for "divisible by 5" and another for "divisible by 10". Every multiple of 10 is already a multiple of 5, so adding counts those cases twice.
• Stopping at 256/625. That is the chance that no number is a multiple of 5, the complement of the event asked.
• Writing odds, such as 256/369, where a probability is asked. A probability divides the favourable cases by all cases, not by the unfavourable ones.

UnlockIAS will link similar questions from RAS Pre 2021 here once that paper is published on this site.

• practice — not a real PYQ

  Three numbers are chosen so that the last digit of each is equally likely to be any digit from 0 to 9, independently of the others. What is the probability that their product is divisible by 5?

  • (a)64/125
  • (b)61/125
  • (c)48/125
  • (d)3/5
  Answer(2) — P(no number is a multiple of 5) = (4/5)³ = 64/125, so P(product divisible by 5) = 1 − 64/125 = 61/125. Option (1) is the complement; option (3) is the chance that exactly one number is a multiple of 5; option (4) adds 1/5 three times, which counts overlapping cases more than once.
• practice — not a real PYQ

  A fair die is thrown twice. What is the probability that the product of the two numbers is even?

  • (a)1/4
  • (b)1/2
  • (c)3/4
  • (d)2/3
  Answer(3) — The product is odd only if both numbers are odd: 1/2 × 1/2 = 1/4. So P(even) = 1 − 1/4 = 3/4. Option (1) is the chance of an odd product; options (2) and (4) do not equal 1 − 1/4.