Types Of Events in Probablity by Pranitaa Shetty (2013042) GROUP 7.

INDEPENDENT EVENT PROPERTIES

Definition:

Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Some other examples of independent events are:

• Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
• Choosing a marble from a jar AND landing on heads after tossing a coin.
• Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
• Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.

A card is  chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and then an eight?
P(jack)  =   4  52 P(8)  =   4  52 P(jack and 8)  =  P(jack)  ·  P(8)  =   4   ·   4  52 52  =    16   2704  =    1   169

DEPENDENT EVENT PROPERTIES

Definition:

Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed.

Now that we have accounted for the fact that there is no replacement, we can find the probability of the dependent events in Experiment 1 by multiplying the probabilities of each event.

Experiment 1:		A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack?
Probabilities:	P(queen on first pick)  =   4  52 P(jack on 2nd pick given queen on 1st pick)  =   4  51 P(queen and jack)  =   4   ·   4   =    16    =    4   52 51 2652 663

MUTUALLY EXCLUSIVE EVENTS

Two events are mutually exclusive if they cannot occur at the same time.

Examples

• When tossing a fair coin, the event 'getting a head' and the event 'getting a tail' are mutually exclusive because they can't occur at the same time.
• When throwing a fair die, the event 'getting a 1' and the event 'getting a 4' are mutually exclusive because they can't occur at the same time. But the event 'getting a 3' and the event 'getting an odd number' are not mutually exclusive since it can happen at the same time (i.e. if you get 3)

For two mutually exclusive events, A and B, the probability of either one occurring, P(A or B), is the sum of the probability of each event.

P(A or B) = P(A) + P(B)

For example, when choosing a ball at random from a bag containing 3 blue balls, 2 green bals, and 5 red balls, the probability of getting a blue or red ball is

P(Blue or Red) = P(Blue) + P(Red)

P(Blue or Red) = ³/₁₀ + ⁵/₁₀

P(Blue or Red) = ⁸/₁₀ = 0.8

This is so because a card can either be red, king, or both (i.e. red king). So that's why we need to subtract the probability of a card being both red and king because it has already been accounted for in the probability of the card being red and the probability of the card being king.