4.2b Probability Rules: Independence, Multiplication Rules, and Conditional Probability

Choose two cards from a standard deck, with replacement. What is the probability of choosing a two and then a five?

Because the cards are replaced after each draw, the two events are independent. Using the Multiplication Rule for Probability, we have. P(two and five) = P(two)P(five) P(two and five) = (4/52)(4/52)≈0.0059.

with repetition

The phrase means that outcomes may be repeated. For example, if you are choosing numbers with repetition for a bank pin number then you are allowed to repeat numbers, such as in the pin 1231.

Conditional probability is necessary in order to find the probability of two events both happening that are not independent, which we will call

dependent

without repetition

means that outcomes may not be repeated. For example, if you are choosing a bank pin without repetition then you are not allowed to repeat numbers, such as in the pin 1234.

When two events are independent, the probability that both events occur can easily be calculated by

multiplying their respective probabilities together.

Multiplication Rules for Probability

the probability of two events both happening, instead of the probability of one or the other happening For example, what is the probability of choosing a queen and then a king from a deck of cards?

The key word in these problems is the word

"and"

In the example of conditional probability: how many kings are available in the deck for the second draw.

One king has been taken out, so there are three kings left. How many cards do we have left in the deck from which to choose? One card was removed, so there are now just 51 cards in the deck. Thus, P(2nd king ∣ 1st king)= 3/51≈0.0588 divide the number of kings left by the total number of cards left

conditional probability can be denoted

P(2nd king ∣ 1st king) This notation is read "the probability of drawing a king second, given that a king was drawn first."

Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a heart for the second card drawn, if the first card, drawn without replacement, was a heart? Express your answer as a fraction or a decimal number rounded to four decimal places.

P(heart∣∣heart)=12/51=4/17≈0.2353 you do not need the first probability - only the probability of the 2nd draw is necessary

Multiplication Rule for Dependent Events

Suppose we wanted to calculate the probability of choosing two face cards in a row, without replacement. For these types of problems, we multiply the probability of one event (choosing 1 face card: P*E) by the conditional probability of the two events (choosing 1 face card and then the 2nd: P(F | E) as shown in the formula:

What is the probability of choosing two non-face cards in a row? Assume that the cards are chosen without replacement.

We are dealing with dependent events, so we must use the Multiplication Rule for Dependent Events. When the first card is picked, all 40 non-face cards are available out of 52 cards. When the second card is drawn, there are only 39 non-face cards left out of 51 cards left in the deck. Thus we have:

without replacement

[a dependent trial] means your first choice is not put back in for consideration, such as drawing a card and then drawing a second card from those left over.

two events are independent if

one event happening does not influence the probability of the other event happening. For example, if after drawing a card you replace the card drawn - and shuffle the deck - then the probability of the next card drawn is not affected by what was picked first; therefore, choosing the two cards from a deck with replacement are independent events.

with replacement

the phrase refers to placing objects back into consideration, such as choosing a card from a deck and then returning it to the deck for the next choice.

with replacement example:

the probability of 3 tails in 3 coin flips each flip is totally independent of what happened before: P(TTT)=P(T)xP(T)xP(T) what is the probability of flipping tails in a coin toss? 1/2 or 50% probability P(TTT)=1/2x1/2x1/2 = 1/8 (multiply numerator and denominators)

When two events are not independent, the outcome of one influences

the probability of the other. For example, consider drawing two cards from a standard deck without replacement. If a king is drawn first, what is the probability that you will draw a king again on your second pick? This is called a conditional probability

example of without replacement

when you draw a red card and don't put it back, the probability of the next card also being red 1-outcome has been removed

Out of 350 applicants for a job, 160 are female and 54 are female and have a graduate degree.

when you see the word "and" automatically think "multiply"

These four phrases affect the number of possible outcomes in the sample space:

with repetition with replacement without repetition without replacement

Out of 350 applicants for a job, 160 are female and 54 are female and have a graduate degree. If 104 of the applicants have graduate degrees, what is the probability that a randomly chosen applicant is female, given that the applicant has a graduate degree? Express your answer as a fraction or a decimal rounded to four decimal places.

you'll need to find the total grad degrees and divide it by the total female portion with grad degrees. *when dividing fractions, flip the numerator with the denominator