Types of Probability

Einstein Was Wrong

Identify each of the following as experiments involving dependent in independent events.

(a) Toss a coin and roll a die six.

(b) Draw 2 cards from a 52 card deck, without replacement.

(c) Draw 2 cards from a 52 card deck, with replacement.

INDEPENDENT DEPENDENT

INDEPENDENT

DEPENDENT

INDEPENDENT

Two marbles are drawn. If the 1st marble is blue it is discarded and replaced with a red. Similarly, if the 1st marble is red it is discarded and replaced with a blue one. What is the probability that the second marble is blue?

Breakfast for RupertRupert has either milk or cocoa to drink for breakfast with either oatmeal or pancakes. If he drinks milk, then the probability that he is having pancakes with the milk is 2/5. The probability that he drinks cocoa is 1/4. If he drinks cocoa, the probability of him having pancakes is 3/8.

a) Show the sample space of probabilities using a tree diagram or any other method of your choice.

b) Find the probability that Rupert will have oatmeal with cocoa tomorrow morning.

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and then replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Independent EventsEvents in which the outcome of one event does not affect the outcome of the other event.

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and NOT replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Dependent EventsIf the outcome of one event affects the outcome of another event, then the events are said to be dependent events.

The probability that Gallant Fox will win the first race is 2/5 and that Nashau will win the second race is 1/3.

3. What is the probability that at least one horse will win a race?

The probability that Gallant Fox will win the first race is 2/5 and that Nashau will win the second race is 1/3.

3. What is the probability that at least one horse will win a race?

2. What is the probability that both horses will lose their respective races?

1. What is the probability that both horses will win their respective races?

Mutually Exclusive Events ...

Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together.

Formally, two events A and B are mutually exclusive if and only if

Examples:1. Experiment: Rolling a die once Sample space S = {1,2,3,4,5,6} Events A = 'observe an odd number' = {1,3,5} B = 'observe an even number' = {2,4,6} A ∩ B = ∅ (the empty set), so A and B are mutually exclusive.

2. A subject in a study cannot be both male and female, nor can they be aged 20 and 30. A subject could however be both male and 20, or both female and 30.

Mutually Exclusive Not Mutually Exclusive

ExampleSuppose we wish to find the probability of drawing either a king or a spade in a single draw from a pack of 52 playing cards.

We define the events A = 'draw a king' and B = 'draw a spade'Since there are 4 kings in the pack and 13 spades, but 1 card is both a king and a spade, we have:

P(A U B) = P(A) + P(B) - P(A ∩ B) = 4/52 + 13/52 - 1/52 = 16/52

So, the probability of drawing either a king or a spade is 16/52 = 4/13.