# 10. A 6-sides die is rolled three times.Part a: Construct the probability distribution table for the number of times the die will land on 6.Part b: Determine the expected number of times the die should land on 6.

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10. A 6-sides die is rolled three times.
Part a: Construct the probability distribution table for the number of times the die will land on 6.
Part b: Determine the expected number of times the die should land on 6.

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Step 1

a.

Conditions to construct a probability distribution table:

Discrete random variable:

If the variable X takes finite or countable values, then the variable X is said to be discrete.

Conditions for the discrete probability distribution:

• The probabilities of all the values of the variable X must lie between 0 and 1.
• The sum of probabilities of all the values of X must be equal to “1”.
Step 2

Construct the probability distribution table for the number of times the die will land on 6.

Consider that the variable X represents the number of times the die will land on 6.

Here, it is given that the six sided die is rolled for 3 times. Therefore, the die can show 6 for 0 times, 1 time, 2 times and 3 times.

That is, the variable X takes the values 0, 1, 2 and 3.

The numbers on the six sided die are 1, 2, 3, 4, 5 and 6.

Since, the die is unbiased and total number of outcomes is 6, the probability of getting 6 is p = 1/6.

The probability of not getting 6 is 1 – p = (1-(1/6)) = 5/6

Here, the die is rolled for three times. This says that the outcome of first roll, second roll and third roll is independent.

No six:

The probability of not getting 6 in any of the three rolls is the product of probabilities of getting no six in the first roll, second roll and third roll.

One six:

The probability of getting one 6 when the die is rolled for 3 times is the product of probabilities of getting one six in the first roll and no six in second and third rolls or the product of probabilities of getting one six in second roll and no six in third and first rolls or the product of probabilities of getting one six in third roll and no six in second and first rolls.

Two six:

The probability of getting two 6 when the die is rolled for 3 times is the product of probabilities of getting one six in the first roll and one six in second roll and no six in third roll or the product of probabilities of getting one six in the second roll and one six in third roll and no six in first roll or the product of probabilities of getting one six in the first roll and one six in third roll and no six in second roll.

Three six:

The probability of getting three 6 when the die is rolled for 3 times is the product of probabilities of getting six in the first roll, second roll and third roll.

The probability distribution table for the number of times the die will land on 6 is obtained below:

Step 3

Find the expected number of times the die should land on 6.

The general formula to obtain the expected value of a discrete random variable is,

E(X) = Summation of all (X*P(X))

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