Welcome to lesson on the probability of

independent events. Let’s begin by reviewing basic probability. The probability of an event represents the likelihood it will occur. Probability compares the favorable number of outcomes to the total number of outcomes. So we

say the probability of event E is equal to this quotient here, where the probability can be expressed as

a fraction, decimal, or percentage. But in this lesson

we’re concerned about independent events, where events A and B are independent if the probability of event B occurring is the same whether

or not event A occurs. So here are two examples of independent events. Number one: a fair die is rolled two times. The outcome of the second role was not

affected by the outcome of the first roll. So we can say the probability of rolling a particular number the second time is the same, whether or not the first roll occurs. Number two: picking a card

from a deck and flipping a coin would also be two independent events. Because the outcome of one event does not affect the outcome of the other. And if two events are not independent they would be dependent. Here’s an example of two dependent events. Draw a card from a deck without replacement and then draw another card. Notice in this case, the probability of selecting a

particular card the second time would be affected by the card that was

drawn the first time. And that’s because there’s no replacement. If there was replacement, then we would

have two independent events. When two events are independent, the

probability have both occurring is the product of the probabilities of the individual events. So again, if events A and B are independent, then the probability of A and B is equal to the probability of, A times the probability of B. Let’s take a look

at some examples. You flip a fair coin and pick one card out of a hat containing twenty cards numbered from one to twenty. What is the

probability of getting heads on the coin and a number greater

than fifteen from the hat? So we want to know the probability of flipping the coin and getting ahead, and picking a number that’s greater than fifteen from the hat. So we’ll say and. And is greater than fifteen. Because these two events are independent this probability is equal to the probability of flipping a head, times the probability of

something a number greater than fifteen from the hat. The probability of getting a heads from the coin flip would be one-half. There’s two outcomes, one of which is

favorable, times the probability of selecting a number

greater than fifteen. There are a total of twenty numbers, there

are twenty possible outcomes. To determine how many favorable outcomes there are, notice how there are five numbers greater than fifteen, sixteen through twenty. So this would be five-twentieths, which does simplify to one-half times, common factor of five, that would be one-fourth, which is equal to one-eighth. So we can say the probability of flipping a head and selecting a number greater than fifteen is equal to one- eighth as a fraction. As a decimal that would be zero point one, two, five, which we can get by taking one and dividing by eight. To bring to a percentage, we’d multiply by one hundred and add a percent sign. So it’s just the same as twelve point five percent. Here’s three different ways to represent this probability. Next, you roll a fair die twice. What is the probability of rolling a three on the first roll and an even number on the second roll? So we want to find the probability of rolling a three, and an even on the second roll? Because these two events are independent, this is equal to the probability of rolling a three, times the probability of

rolling an even number. So the probability of rolling a three, is only one favorable outcome out of six, so that would be one-sixth, times the probability of rolling an even

number. Well two, four and six are even, so there’s three favorable outcomes out of a total of six. Simplify three-sixths to one-half. So one-sixth times one-half is equal to one-twelfth. So we can say the probability of rolling a three and on the second role, rolling an even number, is equal to one-twelfth as a fraction. As a decimal, we have point zero, eight, three repeating, which we’d write like this. If we went to round, we may round two to three decimal places, which would be approximately zero point zero, eight, three. As a percentage, the exact value would be eight point three, with a bar, percent, or we might just say eight point three percent. Let’s take a look at one more example. A

card is pulled from a deck of cards and noted. The card is then replaced, the deck is shuffled, and a second card is removed and noted. What is the probability that both cards are face cards? Well there are a total of fifty-two cards pictured here, where twelve of them: jacks, queens and kings are the face cards, one of each suit, and there are four suits. So I want the probability of a face card with replacement, and then getting another face card the second time. So because there’s

replacement, these two events are independent. If there

wasn’t replacement, they would not be independent. But because we’re independant, this is equal to the probability of a face card, times the probability of another face card. Again there are twelve face cards out of a

total of fifty-two cards, so this would be twelve divided by fifty-two, times twelve divided by fifty-two. Twelve and fifty-two do share a common factor of four. So both of these factions simplify to three-thirteenths. So we have three-thirteenths times three-thirteenths which is equal to nine divided by one hundred sixty-nine. Let’s go ahead and express this again three ways. Probability of a face card and a face card is equal to, as a fraction, nine over one sixty-nine. As a decimal we’d have nine divided by one hundred sixty-nine, let’s go ahead and just round this to four

decimal places. We’ll say it’s approximately point zero five, three, three. Where as a percent, it would be approximately five point three, three percent. Okay, that’s going to do it for this lesson. I hope you found this helpful.

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