What Are The Multiplication Rules For Probability - Dependent And Independent Events in Statistics

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In this video we discuss what are the multiplication rules for probability and how to use them. We cover the two multiplication rules by going through examples.

Transcript/notes
Multiplication rules of probability
When discussing the 2 multiplication rules for probability, we first need to start with defining independent and dependent events.

Independent events are when the outcome of one event does not affect the probability outcome of the other event. For instance, event A is flipping a coin and event B is rolling a die. Whatever happens in event A, flipping the coin, has no affect on event B, rolling the die.

And dependent events are when the occurrence or outcome of the first event changes the probability of the second event. For example drawing a card from a deck, and not replacing it and drawing another card.

Now to the multiplication rules, and the first one is that if 2 events are independent, the probability of both occurring, probability of A and B equals probability of A times probability of B. So, flipping a coin as event A and rolling a die as event B. The probability of getting a head and a 6? Probability of A equals 1 over 2, and the probability of B equals 1 over 6, multiply them and we get 1 over 12, or 0.083 or 8.3% probability of this outcome. And you can see this if you draw a tree diagram, with head and 6 being 1 of 12 possible outcomes.

As another example let’s say that you are a stats nerd and you have calculated out your own win probabilities for your favorite basketball team in their next 5 games. Game one you give them a 64% win probability, game 2 is at 73%, game 3 is at 41%, game 4 is at 44% and game 5 is at 56%. Discarding injuries and load management issues, these are all independent events, so what is the probability they win the next 5 games?

We will assign games 1 thru 5 as events A thru E, and using our first multiplication rule, probability of A and B and C And D and E equals probability of A times probability of B times probability of C times probability of D times probability of E times. We have 0.64 times 0.73 times 0.41 times 0.44 times 0.56, which equals 0.047 or a 4.7% probability to win 5 in a row.

Now for multiplication rule number 2 and it applies when 2 events are dependent. The probability of A and B equals probability of A times probability of B given that A has already occurred. And in the last part of the equation here this slash does not mean divide, it means, as I just stated, probability of B given that A has already occurred.

To use this rule, let’s say we want to find the probability of drawing 2 hearts from a deck, without replacing after drawing. There are 52 cards in a deck, 13 of them are hearts and we will assign drawing a heart as the first card as event A, and drawing a heart as the second card as event B.

So, using our equation or rule, the probability of A and B equals, probability of A, which in this case is 13 over 52, times probability of B given that A has already occurred which is 12 over 51. It’s 12 over 51 because we are assuming that our first draw was a heart, so there are only 12 hearts remaining and 51 because we have removed a possible card from the deck with our first draw. So, doing the math, we get 156 over 2652, which equals 0.059 or a 5.9% probability of drawing 2 hearts from a full deck.

As another example, let’s say that in a certain zip code, 68% of homes have a Netflix subscription, and of these subscribers, 14% also have a Amazon Prime subscription. Based on this data, what is the probability of randomly selecting a house that has a subscription to both services?

We are going to use the formula for multiplication rule number 2 here. The probability of A and B equals probability of A times probability of B given that A has already occurred. In this situation we will assign event A as a home with a Netflix subscription and event B as a home with an Amazon Prime subscription.

We know that probability of A is 0.68 or 68%, as we were given that data, and we actually are given the second part of the equation as well. The probability of B given that A has already occurred means the probability that a home has a Amazon Prime subscription, given they have a Netflix subscription, which is 0.14 or 14%. So, multiplying these we get 0.095 or a probability of 9.5% that a randomly selected home has both Netflix and Amazon Prime.

Timestamps
0:00 Independent And Dependent Events
0:32 Multiplication Rule Number 1
1:12 Example Problem For Rule 1
2:11 Multiplication Rule Number 2
3:20 Example Problem For Rule 2