Data Interpretation & Probability
Independent Events
Suppose one card is drawn from a standard deck of cards and then another is drawn without replacing the first. What is the probability that both are spades?
If the first card is a spade then the probability of the second card being a spade is 12/51. (since there will be 12 spades left, and a total of 51 cards left).
On the other hand, if the first card is not a spade then the probability of the second card being a spade is 13/51.
Notice the probability of the second card being a spade depends on the outcome of the first draw and therefore the two events are dependent.
However, suppose a card is drawn, replaced in the deck, and then the second card is drawn. Then the probability of the second card is a spade is 13/52, regardless of the outcome of the first drawing, and therefore the two events are independent.
Suppose a single card is drawn from a deck, the events "it is a heart" and the event "it is a club" are mutually exclusive since one cannot happen if the other does. However, the events "it is a heart" and the event "it is a red card" are alternative outcomes of a single draw that are NOT mutually exclusive since they can both happen.
Test For Independent Events:
If A and B are events and P(A\B) = P(A) or P(B\A) = P(B) we say that A and B are independent events. In this case we have
Independent events are never mutually exclusive unless one of the events are impossible. So if events A and B are Independent then
Independence simply means that knowledge of one event does not provide knowledge of the other event.
Mutually exclusive events are automatically dependent, since the occurrence of one event tells us something about the occurrence of the other event. ie. If one occurs the other will not. So if events A and B are mutually exclusive then 
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Example 1:
Toss a coin, and then roll a die. The sample space of this experiment consists of 12 elements. Define the following two events, and determine if they are independent.
Solution:
H = Event of tossing a head
D = Event of rolling a three
| DIE |
1 |
2 |
3 |
4 |
5 |
6 |
|
COIN |
(H,1) |
(H,2) |
(H,3) |
(H,4) |
(H,5) |
(H,6) |
|
(T,1) |
(T,2) |
(T,3) |
(T,4) |
(T,5) |
(T,6) |
There is 6 out of 12 ways to have a Head come up in this experiment.
P(H) = 6/12 = 1/2
There is 2 out of 12 ways to have a three come up in this experiment.
P(D) = 2/12 = 1/6
If H and D are independent then
P(H)*P(D) = 
The two values are equal, therefore the two events are independent.
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Example 2:
Roll a pair of dice.
Let the following be two events and determine if they are independent.
Let M = dice show the same number {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
Let S4 = the sum of the two dice is 4 {(3,1),(2,2),(1,3)}
|
1 |
2 |
3 |
4 |
5 |
6 |
|
|
1 |
(1,1) |
(1,2) |
(1,3) |
(1,4) |
(1,5) |
(1,6) |
|
2 |
(2,1) |
(2,2) |
(2,3) |
(2,4) |
(2,5) |
(2,6) |
|
3 |
(3,1) |
(3,2) |
(3,3) |
(3,4) |
(3,5) |
(3,6) |
|
4 |
(4,1) |
(4,2) |
(4,3) |
(4,4) |
(4,5) |
(4,6) |
|
5 |
(5,1) |
(5,2) |
(5,3) |
(5,4) |
(5,5) |
(5,6) |
|
6 |
(6,1) |
(6,2) |
(6,3) |
(6,4) |
(6,5) |
(6,6) |
If these two events are independent then
Since these values are not equal, these two events are not independent.
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Theorem:
If events E1,E2,...,En are independent events then the intersection of the events is given by:
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Example 3:
An urn contains three white marbles and one black marble. A marble is drawn from the urn and replaced, the urn is shaken, and a marble is drawn again. What is the probability that both marbles drawn are white?
Solution:
Since before the second drawing the original situation was restored by replacing the marble, the two events are independent. Each time, the probability of drawing a white marble is 
So the probability of drawing a white marble both times is 
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Example 4:
Suppose you draw 2 cards from a deck of cards, toss a coin 3 times, and roll 2 dice. What is the probability of drawing 2 hearts, getting exactly 2 heads and rolling a sum of 10 with the dice?
Solution:
Let E=2 hearts
Let F = 2 heads
Let G = sum of 10
Since drawing cards from a deck, tossing a coin, and rolling a sum of 10 are independent events, we use the above formula.
There are C(13,2)/C(52,2) ways to draw two hearts, 3/8 ways to get exactly two heads with two flips of a coin, and 3/10 ways to get a sum of 10 when rolling two dice.
Since we want event E AND event F AND event G to occur, we use the Intersection Rule for Independent Event. As usual with the word AND we multiply.
= .00184
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Example 5:
A concert is planned for an outside event. It is calculated that there is a 10% chance that all the chairs are not delivered on time, and a 12% chance that the singing group will not show up. What is the probability that at least one of these problems occur?
Solution:
Let A be the event the chairs are not delivered on time (.10)
Let B be the event the singing group does not show up (.12)
The probability of at least one of these problems occurring is:
chairs not delivered on time and the singing group DOES show up, or chairs delivered on time and the singing group does NOT show up, or chairs not delivered on time and the singing group does not show up. Summary:=
= (.10)(.88) + (.9)(.12) + (.1)(.12)
= .21
Using the complement
The complement to at least one problem is: one minus no problem.P(At least one problem) = 1 - P(No problem)= 
= 1 - (.9)(.88)
= .21
Answer
A Math teacher allows make up tests and has found that the students probabilities of passing a test on the first, second, and third tries are, .7, .65, .43, respectively. What is the probability of passing the test within three tries?
Answer
The probability that a person in the general public will be farsighted is 64%. The probability that a person will be called for jury duty is 2%. If a person from the general public is selected at random what is the probability that they will not be farsighted nor called for jury duty?
Answer
The probability that a person in the general public will be farsighted is 64%. The probability that a person will be called for jury duty is 2%. If a person from the general public is selected at random what is the probability that at least one of these events occur.
Answer
1. If P(E) = .6 and P(F) = .2, compute 
If E and F are independent
2. If P(E) = .6 and P(F) = .2, compute
If E and F are mutually exclusive
3. The probability that person over the age of 60 will develop cancer is .04 and the probability that she/he will develop heart problems is .02. What is the probability that a patient over 60 will develop:
A) At least one of the diseases
B) Both of the diseases
4. During construction on a new building, a contractor estimates that there is an 8% chance of a material shortage, a 10% chance of a strike, and a 20% chance of delays due to bad weather. What is the probability that at least one of these problems occurs?
5. Smith oil company is currently drilling in three different countries, where it is estimated the chances of finding oil to be 30%, 50%, and 80%, respectively. What is the probability that:
A) Oil will be found at all three sites
B) Oil is found at exactly two sites
6. Three teams are making independent attempts to climb a mountain. If their chances of success are 40%, 45% and 50% respectively, what is the probability that at least one team will succeed?
7. A teacher has found that the probabilities of passing a math test on the first, second, and third tries are .80, .75, and .50 respectively. What are the probabilities of passing the test within
A) Two tries
B) Three tries
8. Use problem #6 from homework C3.3.
A) Are C & D Independent?
B) Are R & C independent?
Answer
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