Question

1. A package contains three candies. Of interest is the number of caramel flavored candies in the package. Based on past experience the probability of 0 caramel candies is 0.11, the probability of 1 caramel candy is 0.22, and the probability of 2 caramel candies is 0.36. Let X be the number of caramel candies in the package.

   Find the probability that X = 3.

1. A package contains three candies. Of interest is the number of caramel flavored candies in the package. Based on past experience the probability of 0 caramel candies is 0.11, the probability of 1 caramel candy is 0.22, and the probability of 2 caramel candies is 0.41. Let X be the number of caramel candies in the package.

   Find the probability that X is at most 2?

1. A package contains three candies. Of interest is the number of caramel flavored candies in the package. Based on past experience the probability of 0 caramel candies is 0.11, the probability of 1 caramel candy is 0.22, and the probability of 2 caramel candies is 0.41. Let X be the number of caramel candies in the package.

   What is the expected value of X?

2. A package contains three candies. Of interest is the number of caramel flavored candies in the package. Based on past experience the probability of 0 caramel candies is 0.11, the probability of 1 caramel candy is 0.22, and the probability of 2 caramel candies is 0.34. Let X be the number of caramel candies in the package.

   What is the standard deviation of X?

3. A package contains three candies. Of interest is the number of caramel flavored candies in the package. Based on past experience the probability of 0 caramel candies is 0.11, the probability of 1 caramel candy is 0.22, and the probability of 2 caramel candies is 0.37. Let X be the number of caramel candies in the package.

   Suppose that two random independent packages are selected. What is the probability that there a total of 6 caramel candies?

4. The quality assurance engineer of televisions inspects TV’s from a large factory. She randomly selects a sample of 18 TV’s from the factory to inspect. Assume that 33% of the TV’s in the lot are silver. Let Y be the number of silver TV’s selected.

   Find the probability that at least 6 of the TV’s selected are silver.

1. The quality assurance engineer of televisions inspects TV’s from a large factory. She randomly selects a sample of 18 TV’s from the factory to inspect. Assume that 37% of the TV’s in the lot are silver. Let Y be the number of silver TV’s selected.

   Find the probability that less than 4 of the TV’s selected are silver.

1. The quality assurance engineer of televisions inspects TV’s from a large factory. She randomly selects a sample of 18 TV’s from the factory to inspect. Assume that 31% of the TV’s in the lot are silver. Let Y be the number of silver TV’s selected.

   Find the probability that between 3.5 and 7.5 of the TV’s selected are silver.

1. The quality assurance engineer of televisions inspects TV’s from a large factory. She randomly selects a sample of 18 TV’s from the factory to inspect. Assume that 33% of the TV’s in the lot are silver. Let Y be the number of silver TV’s selected.

   Find the mean of Y.

2. The quality assurance engineer of televisions inspects TV’s from a large factory. She randomly selects a sample of 18 TV’s from the factory to inspect. Assume that 35% of the TV’s in the lot are silver. Let Y be the number of silver TV’s selected.

   Find the standard deviation of Y.

Answer 1

Solution :

1) A random variable X represents the number of caramel candies in the package. X can take four values 0, 1, 2, 3.

We have, P(X = 0) = 0.11, P(X = 1) = 0.22, P(X = 2) = 0.36

We have to find P(X = 3).

Total probability must be equal to one.

i.e. P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1

0.11 + 0.22 + 0.36 + P(X = 3) = 1

0.69 + P(X = 3) = 1

P(X = 3) = 1 - 0.69

P(X = 3) = 0.31

The probability that X = 3 is 0.31.

2) A random variable X represents the number of caramel candies in the package. X can take four values 0, 1, 2, 3.

We have, P(X = 0) = 0.11, P(X = 1) = 0.22, P(X = 2) = 0.41

We have to find P(X = at most 2).

P(X = at most 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X = at most 2) = 0.11 + 0.22 + 0.41

P(X = at most 2) = 0.74

The probability that X is at most 2 is 0.74.

3) A random variable X represents the number of caramel candies in the package. X can take four values 0, 1, 2, 3.

We have, P(X = 0) = 0.11, P(X = 1) = 0.22, P(X = 2) = 0.41

P(X = 3) = 1 - (0.11 + 0.22 + 0.41)

P(X = 3) = 1 - 0.74

P(X = 3) = 0.26

Now the expected value of X is given by,

E(X) = Σ x. P(X = x)

E(X) = (0 × 0.11) + (1 × 0.22) + (2 × 0.41) + (3 × 0.26)

E(X) = 0 + 0.22 + 0.82 + 0.78

E(X) = 1.82

The expected value of X is 1.82.