Probability

1.Define and characterize probability.

Solution:

a) Standard probability definition

Let a random event meet following conditions:
  • number of the events is finite
  • all events have the same chance to occur
  • no two events can occur in the same time


Probability of an event A equals pravdepodobnost1a, n = # of all possible events, m = number of cases favorable for the event A

Stands: 0 ≤ P(A) ≤ 1
Probability of an impossible event : P(A) = 0
Probability of a sure event: P(A) = 1 

b) Conditional probability of some event A, given the occurrence of some other event B:

pravdepodobnost1b

c) Probability of two independent events:

P(A∩B) = P(A)P(B)

d) Probability of two mutually exclusive events:

P(AUB) = P(A) + P(B)

e) Binomial probability expression:

Let an event A occur with probability P. The probability of k occurrances of the event A in n tries equals:

pravdepodobnost1e

f) Hypergeometric probability expression: 

Let V of N elements have a property p; derived from that, N-V elements do not have the property p.
Probability of k of n randomly selected elements having the property p equals::
pravdepodobnost1f

2.There are 18 tickets marked with numbers 1 to 18. What‘s the probability of selecting a ticket having the following property:

a) even number   
b) number divisible by 3
c) prime number   
d) number divisible by 6
Solution:
pravdepodobnost2

3.Determine the probability of following results when throwing 2 playing cubes (a red one and a blue one):

a) sum equals to 8   
b) sum divisible by 5  
c) even sum
Solution:
pravdepodobnost3

4.A gambler playing with 3 playing cubes wants to know weather to bet on sum 11 or 12. Which of the sums will occur more probably?

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5.82 170 of 100 000 children live 40 years and 37 930 of 100 000 children live 70 years. Determine the probability of a 40 years old person to live 70 years.

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6.In a town there are 4 crossroads with trafic lights. Each trafic light opens or closes the traffic with the same probability of 0.5. Determine the probability of:

a) a car crossing the first crossroad without stopping   
b) a car crossing first two crossroads without stopping  
c) a car crossing all the crossroads (4) without stopping  
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7.32 playing cards include 4 aces and 12 figures. Determine the probability of a randomly selected card to be an ace or a figure.

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8.Determine the probability of 3 of 5 born children being sons if the probability of a children to be a boy equals P(A) = 0,51.

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9. There are 16 cola bottles on the table. 10 of them are filled by Coca Cola and 6 of them are filled by Pepsi. Determine the probability of 4 randomly selected bottles to include 2 Coca Cola and 2 Pepsi bottles.

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10.In a game of chance 6 of 49 numbers are the winning numbers. Determine the probability of reaching:

a) matching 3 of 6  
b) matching 4 of 6
c) matching 5 of 6
d) matching 6 of 6
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