Compound events, dependence, independence, and mutually exclusive events.

1. Types of Probability
Einstein Was Wrong

2. Identify each of the following as experiments involving dependent in
independent events.
(a) Toss a coin and roll a die six.
INDEPENDENT
(b) Draw 2 cards from a 52 card deck, without replacement.
DEPENDENT
(c) Draw 2 cards from a 52 card deck, with replacement.
INDEPENDENT
INDEPENDENT DEPENDENT

3. 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?

4. Breakfast for Rupert
Rupert 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.

5. Independent Events
Events 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
then replaced back in the bag. A second marble is selected, what is the
probability that it is blue?

6. Dependent Events
If the outcome of one event affects the outcome of another event,
then the events are said to be dependent events.
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?

7. 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?

8. The probability that Gallant Fox will win the first race is 2/5 and that Nashau
will win the second race is 1/3.
1. What is the probability that both horses will win their respective
races?
2. What is the probability that both horses will lose their respective
races?
3. What is the probability that at least one horse will win a race?

9. 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
Mutually Exclusive Not Mutually Exclusive
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.

10. Example
Suppose 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.