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### Course: High school statistics>Unit 6

Lesson 2: Multiplication rule for probabilities

# General multiplication rule example: dependent events

We can use the general multiplication rule to find the probability that two events both occur when the events are not independent. Created by Sal Khan.

## Want to join the conversation?

• I have another question. What is the possibility that Doug doesn't draw silk?
• The key to this question is finding the probability that Doug draws silk.

In order for Doug to get silk, Maya first has to not get silk (5/6 chance) then Doug has to draw silk (1/5 chance).

So this means that Doug has a (5/6)*(1/5) or 1/6 chance of drawing silk. If he has a 1/6 chance of drawing silk, then that means that he has a 5/6 chance of not drawing silk.
• What is the difference between mutually exclusive event and the independent event?
(1 vote)
• Two events are independent if the occurrence of either event doesn't affect the probability of the other. Coin tosses are independent because a coin has no memory of previous flips; each toss has 50% chance of heads, no matter the previous results.

Two events are mutually exclusive if only one of them can occur. If I toss a coin, the events 'heads' and 'tails' are mutually exclusive, because they cannot both occur on the same toss.
• How do we know when to use the ind or dep of the general multiplication rule formula in a problem? How do we distinguish?
(1 vote)
• Independent events are two events in which the outcomes do not affect each other. Examples include flipping a coin.

Dependent events are two events in which the first event that occurred affects the outcome of the second event. Examples include drawing names out of a hat, without replacement.

In independent events, you use the multiplication rule with the same probability for the second event as when you started. For example, with flipping a coin, the probability of getting heads is 1/2, and the probability of getting tails is the same as that. So, the probability of flipping heads and then tails is 1/2 x 1/2, or 1/4.

For dependent events, you modify the probability of the second event to accommodate what happened in the first one. For example, if there are 10 different names in the hat and you draw one name (probability of 1/10), and don't replace it, there are nine names left in the hat. Now the probability of getting another name is 1/9. So, the probability of getting those certain names is 1/10 x 1/9, or 1/90.

Hope this helps!!
• this is almost the same video as :General multiplication rule example: independent events
(1 vote)
• The words that was in the sentence.

## Video transcript

- [Instructor] We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them will randomly select a card without replacement that will reveal what the star material must be in their craft. Here are the available cards. So I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probably that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying, what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk. Given Maya no silk. This line right over, this vertical line, this is shorthand for given. And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk. She picked first there's six options out of here. Five of them are not silk, so it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them, and four of them are not silk. There's still silk as an option. And it's important to recognize that the probability that Doug gets no silk is dependent on whether Maya got silk or not. So it's very important to have this given right over here. If these were independent events, if Maya picked and then put her card back in and then Doug were to pick separately, then the probability that Doug gets no silk, given that Maya got no silk, would be the same thing, as a probability that Doug gets no silk regardless of what Maya was doing. And so this will end up becoming four over six which is the same thing as two thirds.