Question

Part 4: Probability

In this part, you will work with probabilities.

1) Consider rolling two four sided dice. One die is blue and the other is green.

(a) State the sample space.

(b) Compute the probability of getting two even numbers.

(c) Compute the probability of getting an even sum if the blue die rolls a 2.

2) What is the probability of drawing a four or a heart when drawing a single card from a standard

deck of cards.

3) Consider a machine that lls 275 gallon tanks of water. The owner discovers that the machine does

not always dispense exactly 275 gallons. Instead, they discover that it lls between 272 and 275

gallons. 25% of the time it will ll between 272 and 273 gallons of water, 20% of the time it lls

between 273 and 274 gallons, and 55% of the time it lls between 274 and 275 gallons.

(a) What is the random variable? Is this discrete or continuous?

(b) If the variable is discrete, state the sample space. If the variable is continuous, give the density

curve.

(c) What is the probability that a 275 gallon tank gets less than 272.5 gallons or more than 273.8

gallons of water?

4) The lowest dose of an antibiotic works 89% of the time. 5% of the time, a stronger dose is needed.

In the remaining case, the bacteria is said to be resistant to the antibiotic. Suppose three patients

are given the antibiotic.

(a) What is the probability that at least two patients will need a stronger dose?

(b) What is the probability that the bacteria is resistant in all three patients?

5) Mammography is used as a form of early detection of breast cancer in women. We know that if a

woman has breast cancer, mammography will detect it 90% of the time. If the woman does not have

breast cancer, then mammography will give a positive result 7% of the time. We also know that

there is a 0.8% chance that any given woman has breast cancer.

What is the probability that a woman has breast cancer if she has a positive mammogram?

Answer 1

1a)

Folloiwng is the sample space when two 4 sided are rolled:

In each outcome (a,b), a shows outcome of blue die and b shows outcome of green die.

b)

Out of 16 outcomes, following shows two even outcomes:

So,

P(both even) = 4/16 = 0.25

c)

There are 4 outcomes in which blue die shows 2. Following are the outcomes in which blue die shows 2 and sum is even

So

P(sum even | blue die shows 2) = 2 /4 = 0.50