2. Introduction into the Mathematical Methods

Probability (2/2)

We have just been considering the probability of an event, or a combination of events occurring. Often, however, we want to know the probability of an event occurring given that some other event has already happened. This is called conditional probability.

The conditional probability $Pr\left\{B|A\right\}$ is the probability of B occurring given that A has occurred. The conditional probability can be found from the equation:

In this equation, $Pr\left\{B\cap A\right\}$ is the joint probability of A and B occurring, or the probability that both will occur. If these two events are independent, then the joint probability is the product of the two independent probabilities.

Consider our dice. What is the probability of getting a six in the second toss of the dice if we have already tossed a six with the first throw? Now we know that the joint probability or the probability of throwing two sixes is 1/36 and we know that the probability of getting a six in the first throw is 1/6, so that shows that the probabilities of getting a six on the second throw are also 1/6.

Now let us consider a more complex example. A factory is producing two types of bolt, both designed to do the same task, and they want to simplify their production by just producing the one type. One criterion that they should consider is the failure rate in the produced bolts. To assess this, they have taken a sample of bolts of both types and determined how many of them were okay and how many were defective as shown in the Table below.

Convert these to probabilities:

The probabilities in the body of the Table are joint probabilities in the whole sample, thus Pr{Type A n Good} = 0.56. The probabilities under the total row or column are the marginal probabilities. We can see that the Type A bolts made up 80% of the sample and Type B made up 20%, just as we can see that 73% of the bolts are good and 27% are defective. What we really want to know is, 'What is the probability that a bolt is defective if it is of Type A or B?'

From this information alone it would appear that it is best to produce the Type B bolts instead of Type A bolts.

Questions

3. If Pie in the Sky wins the first race, what is the probability that Doughboy will win the second?
   
   Hint: ↓↑

   Question 3 concerns conditional probability, Pr{A|B} or the Probability of A given that B has occurred. The general form of conditional probability is:

   $$Pr\left\{A\right|B\}=\frac{Pr\{A\cap B\}}{Pr\left\{B\right\}}$$

   Where Pr{A∩B} is the probability that both A and B will occur. If the events A and B are independent, then

   $\begin{array}{l}Pr\{A\cap B\}=Pr\left\{A\right\}\cdot Pr\left\{B\right\}\text{and thus}\\ Pr\left\{A\right|B\}=\frac{Pr\left\{A\right\}\cdot Pr\left\{B\right\}}{Pt\left\{B\right\}}=Pr\{A\}\end{array}$
4. What is the Probability that Raddish will win both races?
   
   
5. What is the most likely combination to win?
   
   
6. You want to make more than one bet on winning both races so that you have a probability of 25% or more that one of your combinations wins. What is the fewest number of combinations that you must select to reach 30%?
   
   
7. In a deck of 52 cards, containing four suits, what is the probability of getting a King in the next dealt card?
   
   
8. You are dealt a hand of 5 cards from a 52 card pack. What is the probability of getting an Ace?
   
   
9. Again you are dealt five cards. What is the probability of getting five spades?