Combining Probabilities, Part 1

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“The screencast for 10b. Combining probabilities we would be looking at three topics. The first first independent or dependent events. The second overlapping or non overlapping events and then in separate screencast 10b.

Part to the at least rule. We ll start off talking about the idea of independent or dependent events because they affect how you compute the probability of multiple events. Remember anytime. You are trying to find the probability of a multiple event.

More than one thing happening you multiply and that s exactly what we have here we want to know is the probability of the next four births at a hospital being boys. An independent or dependent event. If it is a dependent event then the outcome of the first birth affects the outcome of the second birth and we know that that actually does not happen whether a boy or a girl is born in the second birth really has no dependence upon what happened in the birth that came before that event. So that means that for our particular example.

We are looking at independent events. Now that we know that this is an independent event. We are going to find the probability that this will occur. We are looking at the probability of there being four boys in separate births.

So that means that we re going to be looking at the probability of the first birth times..

The probability of the second birth times. The probability of the third birth times. The probability of the fourth birth since there are two possible outcomes. A boy or girl and we are looking for a boy.

The probability that the first birth will be a boy 1 2. Because these are independent events. We forget about the first birth when we re computing. The second birth in this particular case.

The probability that the second birth will be a boy also would be 1 2. And the same thing happens for the third and the fourth birth to find the probability that ball. Four will be boys. We need to multiply 1 2.

Times. 1 2. Times 1 2. Times 1 2.

Or 1 2..

Raised to the fourth power. That gives us 1 16. Or six and a quarter percent. If we change that to a decimal and then move the decimal point over two places just to show you the difference between an independent and a dependent event.

I m going to change gears here a little bit for the purpose of illustration let s say we were looking at a deck of cards. Suppose we were looking at the probability of pulling a queen and then another queen. Which means we re going to be pulling two cards from the deck. One after another there was no replacement that means.

I would be pulling a card and let s assume since the game is going to continue that i pulled a queen and then that card is put aside. So now when i pulled a second card. There was one less in the deck to set that up in our problem that means that for the first pull in the complete deck of cards. There were 4 queens that would allow me to continue in the game out of 52 cards.

Again if the game is going to continue let s suppose that i pulled a queen well. Now that means. There are only three queens left in the deck. And since one of the cards has been put aside there are only 51 possible outcomes.

That s an example of a dependent event..

If i compute that probability have 12 divided by 52. Times 51 which gives me a probability of. 00. 01.

Half a percent not a great chance of this happening. That s an example and a comparison between independent and dependent events moving on to our second problem. In screencast 10 b. The question asks is the probability of drawing a face card or a heart an overlapping or non overlapping event find the probability that this will occur overlapping or non overlapping has to do with the characteristics of the items that we re considering in the probability problem for this to be an overlapping event.

There would need to be face cards that are also hearts if it were a non overlapping event that means that all of the face cards would not be hearts or vice versa now if you consider a standard deck of cards you know that there are some hearts that are face cards so this definitely is an overlapping event with that in mind we have to remember when we find the probability of drawing a face card or a heart that we must not double count the first thing that we re going to do is to figure out individually. What is the probability of each one of these two things happening. I know that there are three face cards jack king and queen for each suit and we have four suits so that means there are 12 face cards out of 52 cards. I also know that there are four suits in a standard deck of cards and thirteen of those are hearts.

I win this particular game by poll either a face card or a heart out of the deck of cards in the case of or situations. We add the individual probabilities. However because it s an overlapping event. We have to figure out how many cards would be counted twice.

If we didn t make some adjustment..

Those would be the king. The queen and the jack of hearts three cards that would be double counted. So we want to subtract off the double counting. So that they re only counted.

Once and subtract them from that sum now we end up with 12 plus. 13. Which is 25 minus. 3.

Which is 22 over 52. That is the probability of pulling a face card or a heart. If you wish to turn this into a percent or a decimal then take 22 divide it by 52. The decimal is 423 thousandths.

If we move the decimal point over by multiplying by 100 we end up with 423 to recap be very careful that when you re dealing with an overlapping event that you subtract off the items with the double characteristics. So that they are not counted twice. Remember that we without doing that we would have those three cards as part of this 12 and as part of that 13 so that s why we need this adjustment. Thank you for viewing this webcast ” .


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