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Question

For the next two (02) items that follow :
Four dice are rolled.

What is the probability that getting a total of the numbers on the dice is at least 23 ?

This question was previously asked in
NDA 1 2026 GAT Question Paper (12-Apr-2026)
The correct answer is

5/1296

To find the probability that the total of the numbers on the dice is at least 23 when four dice are rolled, we need to follow these steps:

  • Each standard six-faced die has numbers from 1 to 6. Therefore, the minimum sum on four dice is 4 (1 on each die) and the maximum sum is 24 (6 on each die).
  • We are concerned with the sums of 23 and 24, as these are the only sums that are "at least 23".

Now, let's calculate the total number of possible outcomes when four dice are rolled:

  • Each die has 6 faces, so for four dice, total outcomes are \(6^4\).
  • Calculating this, we have \(6 \times 6 \times 6 \times 6 = 1296\).

Next, we calculate the favorable outcomes for sums 23 and 24:

  • To get a sum of 24: Each die must show a 6. This is only one possible outcome: (6, 6, 6, 6).
  • To get a sum of 23: Three dice must show 6, and one die must show 5. This can happen in 4 ways: (6, 6, 6, 5), (6, 6, 5, 6), (6, 5, 6, 6), (5, 6, 6, 6).
  • Therefore, there are 4 ways to roll a sum of 23.

Adding the number of outcomes for getting sums 23 and 24 gives us 5 favorable outcomes:

  • 1 outcome for a sum of 24 and 4 outcomes for a sum of 23.

Finally, the probability is calculated as the number of favorable outcomes divided by the total number of outcomes:

  • Probability = \(\frac{5}{1296}\).

Thus, the probability that the sum is at least 23 is \(\frac{5}{1296}\), which matches the given correct answer.

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Similar Questions

  1. What is the probability that getting a total of the numbers on the dice is 6 ?
  2. In a sample survey of a village, the probability that a farmer is in debt is 0.60. What is the probability that three randomly selected farmers are all in debt (assume independence of events)?
  3. The probability that a family owns a laptop is 0.68; that it also owns a desktop is 0.56. If the probability that it owns both is 0.48, then what is the probability that a randomly selected family owns a laptop or a desktop?

Important Questions from Probability

  1. Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?

  2. If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?

    A. 2/3

    B. 3/4

    C. 1/4

    D. 1/9
  3. Statements followed by some conclusions are given below.

    Statements:

    1. A bag has 2 white, 3 black, 4 red and 6 green balls.

    2. 1 ball selected at random from the bag.

    Conclusions:

    I. The probability that a black ball is selected is 1/5

    II. The probability that a red ball is selected is 6/15

    Find which of the conclusions logically follows from the given statement

    A. Only conclusion I follows.

    B. Only conclusion II follows.

    C. Both I and II follow.

    D. Neither I nor II follows.

  4. In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?

  5. A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:

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