AP Statistics / Mr. Hansen

Name: ___________KEY__________

Test #3 on Chapter 4 (Dec. 3 and 4, 1998)

Instructions: Raise hand if you have questions. Do not leave your seat until you have finished.

1.(a,b) State the two tests that we learned for checking whether events A and B are independent. Enter them here:

(a) _______P(A Ç B) = P(A) · P(B)___________

(b) _______P(B | A) = P(B)_________________

(c) Recall the conditional probability formula:

P(B | A) = P(A and B) / P(A)

When is this formula true? (Check one.)

x For all events A and B.
Note: Here we are assuming that P(A) ¹ 0.

 

¨ Only if A and B are independent.

(d,e) Assume that we believe (c). Show that (a) and (b) are equivalent tests. (In other words, show that if you assume (a), you can derive (b), and that if you assume (b), you can derive (a).) Use formula (c) as a starting point for both proofs. These two proofs are quite short.

Proof that (a) Þ (b): Start with P(B | A) = P(A and B) / P(A). If we assume (a) is true, our eqn. becomes P(B | A) = P(A) · P(B) / P(A), which simplifies to P(B). By transitivity, P(B | A) = P(B), which is exactly what (b) says.

Proof that (b) Þ (a): Again start with P(B | A) = P(A and B) / P(A). If we assume (b) is true, our eqn. becomes P(B) = P(A Ç B) / P(A), which becomes eqn. (a) when we multiply both sides by P(A).

2. Match each of the following terms to the definition or example that matches most closely. Warning: One of the blanks on the left does not match any of the lettered items on the right. Fill this blank in with the letter "X."

_E_ probability

A. set of possible outcomes

_A_ sample space

B. a histogram of relative frequencies between 0 and 1 (inclusive) adding to 1, or a curve of nonnegative points having total area of 1 under the curve
_B_ probability distribution

C. example: the number of red cards received when 10 cards are dealt from a well-shuffled deck

_F_ conditional probability

D. a numeric quantity that depends on a random process

_X_ disjoint events

E. long-run relative frequency

_D_ random variable

F. probability that may change depending on additional knowledge that is available
_C_ discrete random variable

G. events that do not affect each other’s probability

_H_ continuous random variable

H. example: area of grass that is covered when a water balloon of random size is dropped from a random height

 

Note: For many of the problems in this test, you must give both an exact (algebraic) answer and a numeric (calculator) answer correct to at least three decimal places.

3. In a large city, _2_% of the residents suffer from TB (tuberculosis). [Blank was 2 or 3.] If you poll 100 residents (assume these subjects are selected randomly and independently), compute

(a) the probability that all 40 of the first 40 people you poll are TB-free

Exact answer = __0.9840_________

Numeric answer = __0.4457______

(b) the probability that at least one of the first 40 people you poll has TB

Exact answer = __1 – 0.9840_________

Numeric answer = __0.5543_________

(c) the conditional probability that subject #39 has TB, given that subjects 1 through 38 are TB-free

Exact answer = __0.02 (trick question: selections are independent)_________

4. Four cards are dealt to you from a well-shuffled deck. Compute each of the following.

(a) the probability that you have two pairs (e.g., two jacks and two kings, two deuces and two nines, etc.—but note that the pairs must be of two different card values)

Exact answer = ___13C2 · 4C2 · 4C2 / 52C4________

Numeric answer = ___0.0104_________
(b) the probability that you have four kings

Exact answer = ___1/52C4, or (4/52)(3/51)(2/50)(1/49)___

Numeric answer = ___0.00000369_________

(c) the probability that you have four kings, given that you have sneaked a peek at the first card and have discovered that it is a jack

Exact answer = ____0______

Numeric answer = ___0_________

(d) the probability that you have four kings, given that you have sneaked a peek at the first card and have discovered that it is a king

Exact answer = ___1/52C3, or (3/51)(2/50)(1/49)________

Numeric answer = ___0.0000480_________

5. The New Joisy Lottery is introducing a new numbers game that works as follows: You will choose a 4-digit number (0000 through 9999). If the random number chosen later that day matches your number, you win $_6000___ on a $1 bet. [Blank was either 5000 or 6000.] Compute the mean, variance, and standard deviation of the random variable X that equals your net winnings. (Remember to include the effect of the $1 you have to pay to buy the ticket.)

Solution will be posted after we have covered random variables.

6. Are the events A = "it is raining outside" and B = "the first person you spot outside is using an umbrella" independent? Why or why not? Use some made-up numbers to make your case somewhat convincing.

P(A) = 0.1
P(B) = 0.01
P(B | A) = 0.25 ¹ P(B)
\ A and B are not independent events. (Umbrella usage is more likely if we know that rain is falling.)

7.(a) Calculate the number of unique license plates of the form 4 letters + 3 digits that could be made. The letters can range from AAAA through ZZZZ and can cover everything in between (e.g., XJQW).

26 · 26 · 26 · 26 · 10 · 10 · 10 = 456,976,000

(b) Would this be enough for all the cars in the United States? Yes.

(c) Why would the actual number of possible plates probably be lower than your answer to part (a) if this scheme were ever actually implemented? (This is not a math question—you can omit part (c) if you prefer not to answer it.)

Some of the 4-letter patterns (e.g., MATH) would have to be excluded because they might disturb or upset people. You can probably think of other 4-letter patterns that the motor vehicle authorities would want to exclude.