- Two events are independent if the outcome of one has no
effect on the other.
- Consider rolling a die and tossing a coin. Does the result of the die have any impact on the coin? These events are independent.
- Consider this experiment, what is the probability of getting a 1 and
a tail?
-------+-------
| |
h t
-----+----- -----+----
| | | | | | | | | | | |
1 2 3 4 5 6 1 2 3 4 5 6
P(t) = 1/6
P(h) = 1/2
P(t and 1 ) = 1/12
- If two events A and B are independent, P(A and B) = P(A) P(B)
- If you draw a card from a deck of cards, replace shuffle the deck and draw again.
What is the probability that you draw the same card twice?
They are independent,
P(A) = 1/52
P(B) = 1/52
P(A and B) = 1/2704
What is the probability that you don't draw the same card twice?
P(A) = 1/52
P(B) = 51/52
P(A and B) = 51/2704 = 1.9%
- What is the probability of having three girls in a row?
P(E) = 1/2
P(E and E and E) = 1/2 * 1/2 * 1/2 = 12.5%
- Earthquake probabilities:
If I read this article correctly, the
probability of an earthquake occurring
in this area in the next 50 years is 1/50.
What is the probability of an earthquake
the next 50 years followed by another in the next 50 years?
P(E) = 1/50
P(E and E) = 1/50 × 1/50 = 1/2500
What is the probability that we will not have an earthquake
in the next 100 years.
49/50× 49/50 = 2401/2500 = 96%
- Two events are dependent if the occurrence of one has an effect on the probability of the other.
- If A and B are dependent events P(A and B) = P(A) P(B given that A has occurred)
- Selecting two hearts from a deck.
P(A) = 13/52
P(B) = 12/51
P(A and B) = 13*12/(52*51)
- Conditional probability for dependent events.
- P(B|A) = n(A ∩ B)/n(A)
- Problem 40, page 613
Probability of 1 can of grape juice = 8/20
P(1 can of grape juice given 1 can) = 7/19
P(3rd can of grape juice) = 6/18
P(3 cans of grape juice) = 336/6840 = 14/285
- Problem 42
P(apple) = 6/20 = 3/10
P(grape| apple) = 8/19
P(orange | (grape and apple)) = 4/18 = 2/9
P(apple then grape then orange) = 3/10*8/19*2/9 = 8/285
- Problem 46,
n(Yellow and 7) = 1
n(Yellow) = 3
P(7 | yellow) = 1/3
- Problem 56
P(Positive Mammogram | does not have breast cancer) =
n(A) = 6944+92,256 = 99,200
n(B ∪ A) = 6944
P(B|A) = 6,944/99,200 = .07 or 7%