Extra exercises on probability (not ordered in level of difficulty)
1. Find the probability for more Heads than Tails if you toss a fair coin a)
20 times b) 15 times.
2. In a box lie 5 white and 5 black balls. Find the probability that you will
get the same number of white and black balls if you pick a) 2 b) 8 c) 7 of them
at random (without returning the balls).
3) In a box lie 4 red and 3 green balls. What is the probability that you will
get as many green as red ones if you pick 6 of them (without returning).
4. From a deck of cards you choose four cards at random and put them on
a table. What is the probability that they will a) all be from different suits (i.e.
one spade, one heart, one diamond, one clove), b) that there are three suits on
the table, c) that there are two suits, d) that all four cards are from the same
suit.
5. Toss a die twice. Let event A be that at least one toss gives 1, event
B that at least one toss gives 4. a) Are A and B independent events? b) Let
event C be that the sum is 13. Are A and C independent?
6. We send a message of ten letters. Each letter is a 0 or a 1. The probability
is 0.01 that a 1 is received as a 0 or that a 0 is received as a 1. a) What is
the probability that the message is correctly transmitted? b) If we send ten
messages, what is the chance that at most one is incorrect?
k
Theory: In a Poisson distribution with probability P [ξ = k] = λk! e−λ , the
expected value is λ.
In the exponential distribution with P [η < t] = 1 − e−λt , the expected value
1
is λ .
7. The remaining life for a light-bulb is assumed to have exponential distribution with expected life-time λ1 = 1000 (hours). a) What is the probability
that it “goes black” within 500 hours? b) Find the probability that 5 of 7 bulbs
stop working within 1000 hours.
8) If a stochastic variable η has distribution function F (t), then the value
m such that F (m) = 12 is called the median of η. a) Find the median of a
stochastic variable with exponential distribution (λ). b) For what value on λ
will the mean and the expected value be equal?
9. We have an aeroplane with two propellers. Both must work if the plane
shall not crash. Both propellers have exponential lifetime, one with expected
life 10000 hours, the other with expected 8000 hours. a) What is the chance
that the aircraft will survive a flight of 10 hours?
9b) Same as 9a but the plane can stay in the air with just one propeller.
9c) Let one propeller have parameter λ, and the other µ. Find the distribution function for the remaining time that the plane can stay in air if (i) both
propellers must work, (ii) one propeller is enough.
10. We have a shop where the number of customers in a given minute has
Poisson distribution (λ = 3). a) Find the probability that there are exactly
10 customers during a given five-minute interval. b) What is the chance that
no customer arrives during the next 30 seconds? c) What is the probability
function for the number of customers in one hour?
1
11–13. Let a system consist of two components along the same line. One
works with probability p, the other with probability q. If one component fails,
the system will crash.
11a) Find the probability that the system works.
11b) Same situation but now the probability that the first component crashes
is r, and that the other crashes is s. Express the probability that the system
will crash in r and s.
12ab) Same as 11ab) but now the components are arranged in parallel;
hence unless both components fail, the system will work.
13ab) Generalize 11 and 12 to n components in the system. In particular, what happens to a system with n components in parallel where each has
probability n1 to fail, independent of the others.
Answers (some may be wrong; many are given with absurdly unrealistic
accuracy)
1a) 0,8238 b) 12
2a) 59 b) 59 c) 0
3) 47
4. Let N = 124950; te respective probabilities are 13182/N , 73008/N , 37440/N , 1320/N .
5a) No b) Yes
6a) 0.904382 b) 0.753029
7a) 0.3935 b) 0.2868
8a) lnλ2 b) –
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9a) 0.99775 b) 0,99999875 c) (i) 1 − e(λ+µ)t ; (ii) 1 − e−λt (1 − e−µt )
180k
10a) 0.001936 b) 0.22313 c) p (k) = k!e
180
11–13) Think them over; there is a nice discovery to make:)
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