Question

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?

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?

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.

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.

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.

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.

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) = 0.34

P(X = 3) = 1 - (0.11 + 0.22 + 0.34)

P(X = 3) = 1 - 0.67

P(X = 3) = 0.33

Standard deviation of X would be given by,

SD = √(Var(X))

Where, Var(X) = E(X²) - (E(X))²

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 × 0.34) + (3 × 0.33)

E(X) = 0 + 0.22 + 0.68 + 0.99

E(X) = 1.89

Now,

E(X²) = Σ x². P(X = x)

E(X²) = (0² × 0.11) + (1² × 0.22) + (2² × 0.34) + (3² × 0.33)

E(X²) = 0 + 0.22 + 1.36 + 2.97

E(X²) = 4.55

Hence, Var(X) = 4.55 - (1.89)²

Var(X) = 0.9779

Hence, SD = √(0.9779) = 0.9889

The standard deviation of X is 0.9889.