More of David Palay’sSlides

Conditional Probability

Given that you had a quiz on Tuesday, what is the Probability that you have

homework tonight…

Example:

Suppose one employee is selected from a group of random employees. We’re interested if this employee has used the proper cover sheet for his TPS report (since there’s a new cover sheet and we’re not certain that this employee has read the memo). We could ask:• What is We could ALSO ask,• What is

P(improper TPS∨Male)• This is the probability of the employee’s TPS

report having the wrong coversheet, GIVEN that the employee is male.

Conditional Probability

• Written as • Read asProbability of event B, given event A has already happened.

Our Company

• There are 15 people in the small company.• 9 are male• 5 females filed improper TPS reports• 8 employees filed improper TPS reports.

• From this, we can figure out the probability of having an improper TPS report given that the employee is male.

Method 1

• Create a table:

• Use table to calculate probability

Male Female Total

Proper TPS

Improper TPS

Total

Example

• When rolling 2d6, what is the probability that the sum is a 7, GIVEN THAT ONE OF THE DICE IS A 1.

{1,1} {1,2} {1,3} {1,4} {1,5} {1,6}{2,1} {2,2} {2,3} {2,4} {2,5} {2,6}{3,1} {3,2} {3,3} {3,4} {3,5} {3,6}{4,1} {4,2} {4,3} {4,4} {4,5} {4,6}{5,1} {5,2} {5,3} {5,4} {5,5} {5,6}{6,1} {6,2} {6,3} {6,4} {6,5} {6,6}

Method 2: Calculating

P (B|A )= P (A ∩B )P (A )

TPS Reports

Dependent and Independent events

• Dependent events– 2 events where the outcome of one is influenced

by the outcome of the other. • Independent events– 2 events who’s outcomes are completely

unchanged based on the outcomes of each other.

Dependent and Independent events

• If • Or if • Then the events are independent!

Tossing two Coins

• If A is getting heads on the first toss, and• If B is getting heads on the second toss,Are A and B dependent or independent?

Tossing two Coins

• Find • Find

Tossing two Coins

• Event A: getting Heads on first toss• Event B: getting Heads on second toss• Find

• Find

A useful result!

• Take the mathematical equation of conditional probability:

Multiply both sides by We can calculate with just and . This is the MULTIPLICATION RULE for intersections.

Even more useful!

• If we think about independent events for a minute, we can recall that for two events A and B that are independent:

But that means…

In our Multiplication Rule for Intersections

If we are looking at ONLY independent events where

Then, the Multiplication Rule results in

This works for the probability of more than 2 events!

• If A, B, C, D, … are all INDEPENDENT events (coin flips, single die rolls, etc.), then

Ok, take a second…. Breath…

• We just covered a lot with very little explanation. So, we’re going to do some examples.

Dependent vs. Independent

Determine whether the events are independent or dependent

1) A: a coin comes up heads B: 1d6 comes up even

2) A: 1d6 is thrown and comes up evenB: a second d6 is thrown and the sum of the two is greater than 4

3) A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck

Uh-oh…. Wait a sec.

A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck

We can come up with two different answers. It depends on if we put the first card BACK or NOT!

Replacement

• We talked about it with combinations & permutations, and now we need to incorporate it into independent & dependent events.

Replacement

Namely, if we have replacement, does it make it dependent or independent?

Take our deck of cards example.

A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck

Replacement

If we put the first card back into the deck (and re-shuffle) then we have an independent event. The first card has no impact on what the second card could be.

Replacement

Alternatively, if we leave the first card out, then the probabilities for the second card have changed.

One more thing

• Find

• Find