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Chapter 18: Problem 759

3 dice are tossed. Find the probability that sum of digits is 14 (a) \(\left(21 / 6^{3}\right)\) (b) \(\left(15 / 6^{3}\right)\) (c) \(\left(27 / 6^{3}\right)\) (d) \(\left(16 / 6^{3}\right)\)

Short Answer

Expert verified
The probability that the sum of the digits is 14 when 3 dice are tossed is (b) \(\left(15 / 6^{3}\right)\).

Step by step solution

01

Determine the total number of possible outcomes

When 3 dice are rolled, each die has 6 possible outcomes (since there are 6 faces on each die). To find the total number of possible outcomes, we use the multiplication rule, which states that the number of possible outcomes in a sequence of independent events can be calculated by multiplying together the number of possibilities for each event. In this case, the total number of possible outcomes is \(6^3\).
02

Determine the number of successful outcomes

In order to find the number of successful outcomes, i.e., the total number of outcomes where the sum of the numbers on the dice is equal to 14, we can consider different possible cases: 1. One die shows a 6, the second die shows a 6, and the third die shows a 2. There are 3! = 6 ways to arrange this case. 2. One die shows a 6, the second die shows a 5, and the third die shows a 3. There are 3! = 6 ways to arrange this case. 3. One die shows a 5, the second die shows a 5, and the third die shows a 4. There are 3!/2! = 3 ways to arrange this case (since we have two repeating numbers). So, the total number of successful outcomes is 6 + 6 + 3 = 15.
03

Calculate the probability

Now that we have the total number of possible outcomes (\(6^3 = 216\)) and the total number of successful outcomes (15), we can divide the number of successful outcomes by the total number of possible outcomes to find the probability: Probability = \(\frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}} = \frac{15}{216}\) Therefore, the probability that the sum of the digits is 14 when 3 dice are tossed is: \[\frac{15}{6^3}\] Comparing the result to the given options, we can see that the answer is (b) \(\left(15 / 6^{3}\right)\).

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Most popular questions from this chapter

A dice is loaded so that the probability of face \(\mathrm{i}\) is proportional to i. \(\mathrm{i}=1,2, \ldots .6\). Then the probability of an even number occupy when the dice is rolled is (a) \((2 / 7)\) (b) \((3 / 7)\) (c) \((4 / 7)\) (d) \((5 / 7)\)

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