What do you call events that do not affect the probability of each other?

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.

If two events are disjoint, then the probability of them both occurring at the same time is 0.

   Disjoint:  P(A and B) = 0

If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.

Specific Addition Rule

Only valid when the events are mutually exclusive.

   P(A or B) = P(A) + P(B)

Example 1:

Given: P(A) = 0.20, P(B) = 0.70, A and B are disjoint

I like to use what's called a joint probability distribution. (Since disjoint means nothing in common, joint is what they have in common -- so the values that go on the inside portion of the table are the intersections or "and"s of each pair of events). "Marginal" is another word for totals -- it's called marginal because they appear in the margins.

BB'MarginalA0.000.200.20A'0.700.100.80Marginal0.700.301.00

The values in red are given in the problem. The grand total is always 1.00. The rest of the values are obtained by addition and subtraction.

Non-Mutually Exclusive Events

In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.

General Addition Rule

Always valid.

   P(A or B) = P(A) + P(B) - P(A and B)

Example 2:

Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15

BB'MarginalA0.150.050.20A'0.550.250.80Marginal0.700.301.00

Interpreting the table

Certain things can be determined from the joint probability distribution. Mutually exclusive events will have a probability of zero. All inclusive events will have a zero opposite the intersection. All inclusive means that there is nothing outside of those two events: P(A or B) = 1.

BB'MarginalAA and B are Mutually Exclusive if this value is 0..A'.A and B are All Inclusive if this value is 0.Marginal..1.00

"AND" or Intersections

Independent Events

Two events are independent if the occurrence of one does not change the probability of the other occurring.

An example would be rolling a 2 on a die and flipping a head on a coin. Rolling the 2 does not affect the probability of flipping the head.

If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring.

Specific Multiplication Rule

Only valid for independent events

   P(A and B) = P(A) * P(B)

Example 3:

P(A) = 0.20, P(B) = 0.70, A and B are independent.

BB'MarginalA0.140.060.20A'0.560.240.80Marginal0.700.301.00

The 0.14 is because the probability of A and B is the probability of A times the probability of B or 0.20 * 0.70 = 0.14.

Dependent Events

If the occurrence of one event does affect the probability of the other occurring, then the events are dependent.

Conditional Probability

The probability of event B occurring that event A has already occurred is read "the probability of B given A" and is written: P(B|A)

General Multiplication Rule

Always works.

   P(A and B) = P(A) * P(B|A)

Example 4:

P(A) = 0.20, P(B) = 0.70, P(B|A) = 0.40

A good way to think of P(B|A) is that 40% of A is B. 40% of the 20% which was in event A is 8%, thus the intersection is 0.08.

BB'MarginalA0.080.120.20A'0.620.180.80Marginal0.700.301.00

Independence Revisited

The following four statements are equivalent

1. A and B are independent events
2. P(A and B) = P(A) * P(B)
3. P(A|B) = P(A)
4. P(B|A) = P(B)

The last two are because if two events are independent, the occurrence of one doesn't change the probability of the occurrence of the other. This means that the probability of B occurring, whether A has happened or not, is simply the probability of B occurring.

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Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.

$$Probability=\frac{The\, number\, of\, wanted \, outcomes}{The\, number \,of\, possible\, outcomes}$$

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Example

What is the probability to get a 6 when you roll a die?

A die has 6 sides, 1 side contain the number 6 that give us 1 wanted outcome in 6 possible outcomes.

Independent events: Two events are independent when the outcome of the first event does not influence the outcome of the second event.

When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.

$$P(X \, and \, Y)=P(X)\cdot P(Y)$$

To find the probability of an independent event we are using this rule:

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Example

If one has three dice what is the probability of getting three 4s?

The probability of getting a 4 on one die is 1/6

The probability of getting 3 4s is:

$$P\left ( 4\, and\, 4\, and\, 4 \right )=\frac{1}{6}\cdot \frac{1}{6}\cdot\frac{1}{6}=\frac{1}{216}$$

When the outcome affects the second outcome, which is what we called dependent events.

Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event. The probability of two dependent events is the product of the probability of X and the probability of Y AFTER X occurs.

$$P(X \, and \, Y)=P(X)\cdot P(Y\: after\: x)$$

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Example

What is the probability for you to choose two red cards in a deck of cards?

A deck of cards has 26 black and 26 red cards. The probability of choosing a red card randomly is:

$$P\left ( red \right )=\frac{26}{52}=\frac{1}{2}$$

The probability of choosing a second red card from the deck is now:

$$P\left ( red \right )=\frac{25}{51}$$

The probability:

$$P\left ( 2\,red \right )=\frac{1}{2}\cdot \frac{25}{51}=\frac{25}{102}$$

Two events are mutually exclusive when two events cannot happen at the same time. The probability that one of the mutually exclusive events occur is the sum of their individual probabilities.

$$P(X \, or \, Y)=P(X)+ P(Y)$$

An example of two mutually exclusive events is a wheel of fortune. Let's say you win a bar of chocolate if you end up in a red or a pink field.

What is the probability that the wheel stops at red or pink?

P(red or pink)=P(red)+P(pink)

$$P\left (red \right )=\frac{2}{8}=\frac{1}{4}$$

$$P\left (pink \right )=\frac{1}{8}$$

$$P\left ( red\, or\, pink \right )=\frac{1}{8}+\frac{2}{8}=\frac{3}{8}$$

Inclusive events are events that can happen at the same time. To find the probability of an inclusive event we first add the probabilities of the individual events and then subtract the probability of the two events happening at the same time.

$$P\left (X \, or \, Y \right )=P\left (X \right )+ P\left (Y \right )-P\left (X \, and \, Y \right )$$

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Example

What is the probability of drawing a black card or a ten in a deck of cards?

There are 4 tens in a deck of cards P(10) = 4/52

There are 26 black cards P(black) = 26/52

There are 2 black tens P(black and 10) = 2/52

$$P\left ( black\, or\, ten \right )=\frac{4}{52}+\frac{26}{52}-\frac{2}{52}=\frac{30}{52}-\frac{2}{52}=\frac{28}{52}=\frac{7}{13}$$

What is it called when the probability of an event is not affected by a previous event?

What is a null event in probability?

What is independent and dependent events?