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.