Probabilities I: simple probability

In simple probability, we can define the following elements:

• event: a situation we want to achieve (e.g.: getting an odd number when rolling a die; picking a card of hearts from a deck of cards).

• possible cases: the set of all the likely cases in an experiment (e.g.: the numbers {1,2,3,4,5,6} in the experiment of rolling a die; the set {head, tail} in the experiment of throwing a coin).

• success cases: the set of all the cases where the event is true (e.g.: if the event is getting an odd number when rolling a die, the success cases is the set {1,3,5}; if the event is picking a card of hearts from a deck of cards, the success cases is the set {A of hearts, 2 of hearts, 3 of hearts, 4 of hearts, ... , 12 of hearts}).

In general we say that, if A is an event, then the probability of A happening is the result of dividing the number of success cases by the number of possible cases:

P(A) = s / n

where:

A is the event;

s is the number of success cases;

n is the number of possible cases.

Events are called with the first capital letters: A, B, C, etc.

Properties:

• A probability lies between 0 and 1.
• If an event cannot happen, the probability of it occurring is 0 (for example, getting a 7 when rolling a die of numbers 1 to 6).
• If an event is certain to happen, the probability of it occurring is 1 (for example, getting a total lower than 15 when rolling two dice and adding the results; because in any throw, the sum is at least 2 (1+1) and at most (6+6), and the sum is always less than 15).
• If P(A) = p, the probability of A not occurring is P(not A) = 1-p (for example, if the probability that today will rain is 0.7, then the probability that today will NOT rain is 1-0.7 = 0.3).

Examples:

1) Consider the experiment of rolling a die. What is the probability of getting:

a) a six?

A = {getting a six}
s = 1 (only success case: getting a '6')
n = 6 (6 possible results)

So, P(A) = 1/6

b) a number greater than 3?

B = {getting a number greater than 3}
s = 3 (success cases: {4,5,6})
n = 6 (6 possible results)

So, P(B) = 3/6 = 1/2

c) a three or a five?

C = {getting a 3 or a 5}
s = 2 (success cases: {3,5})
n = 6 (6 possible results)

So, P(B) = 2/6 = 1/3


2) A bag contains 10 red balls, 5 blue balls and 7 green balls. Find the probability of:

a) getting a red ball: 10/22 = 5/11

b) getting a blue or a green ball: 12/22 = 6/11

c) not getting a green ball:

A = {getting a green ball}
P(A) = 7/22

P(not A) = 1 - P(A) = 1 - 7/22 = 15/22

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PRACTICE FROM THE BOOK: Pages 351-353 (Exercise 8)