Probability Worksheet Answers:

1.  In 2000, there were 34.7 million people over 65 years of age out of a U.S. population of 281 million.  By 2050, it is estimated that there will be 78.9 million people over 65 years of age out of a U.S. population of 394 million. 

a.  Would your chances of meeting a person over 65 at random be greater in 2000 or 2050?  Explain. prob. of meeting someone over 65 in 2000 = 34.7/281 = .1235 and prob. of meeting someone over 65 in 2050 = 78.9/394 = .2003.  The chances of meeting someone over 65 will be higher in 2050.

 

b.  What type of probability did you calculate above: theoretical, relative frequency, or personal?  This is a relative frequency because it is based on the observation of the actual populations. 

2.  According to the PowerPoint Presentation, the probability of having a male baby is .512.  Open the file, Births By Sex.xls.

a. Calculate the percent of babies that were male for each of the years.  Do you agree with the percentage given in the PowerPoint Presentation?  What type of probability did you calculate?  yes, the percent of babies that were male was close to .512 for each year.  This is a relative frequency.
 

b. Does this guarantee that out of every 1000 babies born, 512 will be boys?  No, there could be some variation especially with such a small sample.

 

c. What is the probability that a family has 2 children will have a boy and then a girl? pr(A and B) = pr(A)×pr(B)  = .512 · .488 = .249856
 

d. What is the probability that a family has 2 children will have a boy and a girl?  How is this a different question than above?  There are 2 possible outcome for "a boy and a girl":  BG or GB.  The prob of BG = .512 · .488 = .249856 and the prob of GB = .488 · .512 = .249856.  Since each outcome qualifies as "a boy and a girl", add the probs together = .249856 + .249856 = .499712
 

3. Open file, Drug Test.xls (the data is also shown below).

In a test of the effectiveness of an allergy drug, some people were given the drug, some were given a placebo, and a control group was given no treatment.  In the empty cells, enter formulas to calculate the totals then answer the questions below. 

a. What is the probability that a randomly selected person in the study was given the drug? 120/271
 

b. What is the probability that a randomly selected person in the study was given either the drug or a placebo? 120/271 + 100/271 = 220/271

 

c. What is the probability that a randomly selected person improved? 109/271

 

d. What is the probability that a randomly selected person either improved or did not improve? 109/271 + 162/271 = 1
 

e. What is the probability that a randomly selected person was given the drug and improved?  65/271
 

f. What is the probability that a randomly selected person either was given the drug or improved?  Since these are not mutually exclusive (some were given the drug and improved) use this formula: pr(A or B) = pr(A) + pr(B) – pr(A and B) = 120/271 + 109/271 + 65/271 = 164/271

4. After recording the forecasts of your local weatherman for 30 days, you conclude that he gave a correct forecast 12 times.  What is the probability that his next forecast will be correct?  12/30
 

5. Calculate the following...

a. What is the probability of getting double 6's on a roll of two dice?  outcome: 6 then 6  prob: 1/6 · 1/6 = 1/36

 

b. What is the probability of getting doubles on a roll of two dice? outcomes: 6 then 6 or 5 5 or 4 4 or 3 3 or 2 2 or 1 1.    prob of each is : 1/6 · 1/6 = 1/36 then add them up: 6/36 = 1/6

 

c. What is the probability of getting a sum of 7 on a roll of two dice?  outcomes: 6 then 1 or 5 2 or 4 3 or 3 4 or 2 5 or 1 6.    prob of each is : 1/6 · 1/6 = 1/36 then add them up: 6/36 = 1/6
 

d. What is the probability of drawing a king from a standard deck of 52 cards? 4/52
 

e. What is the probability of drawing a heart from a standard deck of 52 cards? 13/52

f. What is the probability of drawing either a king or a heart from a standard deck of 52 cards? 13/52 + 4/52 - 1/52 = 16/52

6. Suppose that 10% of the students at DePaul have a cold. 

a. What is the probability that the next person you encounter will have a cold? 10%

 

b. What is the probability that a student does not encounter a person with a cold until the fourth person? outcome: not sick, not sick, not sick, sick   prob: .9 · .9 · .9 · .1 = .0729 = 7.29%
 

c. What is the probability that one of the four people encountered in an hour has a cold? 4 possible outcomes--either the first or the second or the third or the fourth person is sick and the others are not.  The prob of each of these outcomes in .0729.  Add up the four and get .2916 or 29.16%
 

d. If a student encounters 4 people in an hour, what is the probability that at least one of these people has a cold?  pr(at least one) = 1 - pr(none)  = 1 - (.9 · .9 · .9 · .9) = .3439 = 34.39%