Exercises Probability –Applied statistics
1.
3.
A, B and C are events. Express the following events in A, B and C using union, intersection and
complement of these events (First draw a “general” Venn-diagram for A, B and C).
a. A and B, but not C. b. All three
c. At least out of three d. At least two
e. None
f. Exactly one of three g. Not more than 1
We define for a collection of 1200 screws the following events:
A = the set of screws having a length of (at least) 10 cm.
B = the set of screws having a weight of (at least) 10 g.
C = the set of screws having a diameter of (at least) 10 mm.
400 screws have a length of 10 cm and a weight of 10 g.
400 screws have a length of 10 cm and a diameter of 10 mm.
400 screws have a diameter of 10 mm and a weight of 10 g.
300 screws have a length of 10 cm, a weight of 10 g and a diameter of 10 mm.
Compute the probability that an arbitrarily chosen screw is an element of at least 2 of the 3 events A, B , C.
If P(A)= ½ ,
= ⅓ and
, compute P(B) and
).
4.
Compute
2.
if
= 0.8,
= 0.3 and
= 0.35.
5.
A cupboard has 3 drawers: in the first there are two golden coins, in the second one golden and one silver
coin and in the third two silver coins. Somebody opens an arbitrary drawer and chooses an arbitrary coin: it
is a golden coin. What is the probability that the second coin from this drawer is golden too?
6. A car driver has caused an accident and has to take an alcohol test. Research has shown that 75% of the
persons who had (too much) alcohol shows a positive test result. However, 2% of the persons who did not
drink also show a positive test result. Assume that in case of accidents 5% of the car drivers have been
drinking alcohol. What is the probability that such a driver has a positive test result? (First define events!)
7. We throw a dice nine times. Determine the probability that: a. the first three results are all 5 and the rest
different from 5 and
b. at most 2 throws result in 5 or 6.
8. A student is working at an exam with Multiple Choice Items. Each question has two possible answers (a, b),
one of which is correct. If he does not know the answer, he flips a coin (Tail = a, Head = b). Suppose that
he does not know the answer on 3/5th of the items.
a. What is the probability that he knew the answer to a question that he answered correct?
b. If there are 25 items, what is the probability of at least 23 correct answers?
9. We throw a dice twice (an unbiased dice, independent throws)
a. Give the sample space of this experiment and the probability of every outcome.
b. If Zij X = the sum of both results, determine the outcomes in in the event { X= 7}.
c. Determine the distribution of X.
d. Compute E(X) and σX.
10.
The life expectancy of a machine (in years) can be modelled with the following distribution:
x
P(X= x)
a.
0
0.05
1
0.25
2
0.30
3
0.20
4
0.10
5
0.05
6
0.05
Compute the probability that the machine will not break down during at least 5 years under the
condition that it has survived 3 years
b.
Compute the expected value of the life expectancy
c.
Compute the standard deviation of the life expectancy.
11.
Which of the following problems can be modelled with the binomial distribution? If so, which
parameters?
a.
X is the number of defective machines if we buy 4 arbitrary machines of a manufacturer who has
three defective and 7 good machines in stock.
b.
Compute the probability that at least 9 out of 40 persons respond to a mailing if usually about 15%
respond
c.
A machine is subject to wear so that at the end of the day it produces more failures. We want to
compute the probability of at least one failure if we choose 10 products from a day`s production.
d.
Data from the past that 10% of the clients of car insurance company claim money in a year. A
random sample of 200 clients is sent a questionnaire. X is the number of claims these clients submitted last
year.
12.
X has a B(25, 0.2)-distribution.
Compute a. E(X2)
b. P(X > 2)
13.
The following simultaneous probability function of X and Y is given as follows:
y
x
2
5
6
1
3
6
8
0
0.2
0
0.1
0
0.2
0.1
0
0.1
0
0
0.3
a. Determine the marginal probability function of X and Y .
b. Are X and Y independent?
c. Compute E(X), E(Y ), var(X), var(Y ), cov(X, Y ), ρ(X, Y ).
d. Determine the distribution of 5X + 3Y .
14. For X and Y we have var(X) = 3, var(Y) = 4 and cov(X, Y) = −2.
Compute var(X + Y ) and var(X − 2Y )
15. X ~ N(0, 1), Y ~ N(1, 4) and Z ~ N(μ, σ2). Use the standard normal table to compute:
a. P(X ≤ 3), P(X ≤ −2), P(−2 < X < 3) and P(−2 ≤ X ≤ 3).
b. P(Y ≤ 7) en P(Y ≤ −3).
c. P(μ − σ ≤ Z ≤ μ + σ) d. P((Y − 3) 2 ≤ 16) e. P(5X − Y ≤ 1), if X and Y are independent.
16. The length (in cm) of Dutch men can be considered a N(182, 50)-distributed variable. What is the
percentage of Dutch men having a length between 175 and 185 cm.
17. The diameter of green peas can be modelled as a r.v. X, having a N(6, 2.25)-distribution. Using sieves the
producer of peas wants to sell the peas in 4 classes “large” , “middle” , “fine” and “very fine”. These
classes should be 10, 40, 30 and 20% of the total number of peas respectively. What size should the sieves
have to establish these proportions? (Hint: draw the normal distribution and the proportions
18. The weight (in grams) of an apple is approximately N(150, 400). How many apples (n) should one choose
so that the probability of having more than 2.5 kg of apples is at least 90%.
19. A computer is going to generate 1200 random numbers between 0 and 1. The expectation and variance of
such a number is known: μ = 0.5 and σ2 = 1/12.
a. Compute the expected value and standard deviation of the mean of these 1200 numbers.
b. Compute the probability that the mean is between 0.49 and 0.51, using the Central Limit Theorem.
20. X is B(20, 0.3) and Y is B(1000,0.4).
a. Use a table or a calculator to find the exact value of P( X ≤ 3 ) in 3 decimals.
b. Approximate P( X ≤ 3 ) using the normal distribution with continuity correction. Compare the
result of a and b.
c. Find μY en σY and approximate P( X ≤ 380 ) and P( X < 380 ).
21. A political researcher is conducting a poll to estimate the popularity of party A in a country, that scored
25% during the last elections. He asks at random 250 voters whether they will vote party A when elections
would be organized.: X is the number of party A voters among the 250.
a. What distribution does X have? And which distribution can you use to approximate probabilities for X?
b. What is the probability that the poll predicts a loss of at least 3% (so, at most 22% in the sample) if in the
population the popularity of party A is still the same?