Question

1. Students were introduced to events that were independentand those events that were not independent within the required reading. Two rules appeared within your required reading for this week:

The General Multiplication Rule:  

The Special Multiplication Rule:

(a) State in your own words what it means for two events to be indepedent events.

(b) Provide an original example involving two independent events where you have to find P (A and B). Select the appropriate rule above to compute P (A and B). Identify what A and B represent in your example. No credit will be given if you do not show all work. If applicable, make sure that you simplify your final result.

(c) Provide an original example involving two events that are NOT independent where you have to find P (A and B). Select the appropriate rule above to compute P (A and B). Identify what A and B represent in your example. No credit will be given if you do not show all work. If applicable, make sure that you simplify your final result.

(d) One of the most important topics in probability includes the determination of conditional probabilities. Suppose that we have two events A and B. Explain in your own words when and when

Answer 1

(a)

Two events are said to be independent when the probability of one is not depend on other.

In general if two events A and B are happen then the probability of happening of both the events together is

This is general multiplication rule.

Here, the probability of second event (B) is depend on the probability of first event (A).

But if it is said that the two events A and B are independent then the probability of happening of both the events together is

This is special multiplication rule.

Here, Probability is the result of multiplication of individual probabilities.

(b)

Example of independent events.

Let the experiment consist of drawing a red ball a black with replacement ball out of 2 red balls and 3 black balls.

Let event A denotes the drawing of a red ball.

And event B denotes the drawing of a black ball.

Now, Probability of drawing a red ball is P(R)=2/5

And probability of drawing a back ball is P(B) =3/5

Now, probability of drawing a red ball and a black ball simultaneously when drawing of ball is performed with replacement is given by special multiplication rule as

So, here when second ball is drawn then it does not depend on the first ball because the for second ball all the balls are available as first ball is palced black after it's drawing.

(c)

Example of dependent events.

Let the experiment consist of drawing a red ball then a black without replacement ball out of 2 red balls and 3 black balls.

Let event A denotes the drawing of a red ball.

And event B denotes the drawing of a black ball.

Now, Probability of drawing a red ball is P(R)=2/5

And probability of drawing a back ball is P(B) =3/5

Now, probability of drawing a red ball and then a black ball simultaneously when drawing of ball is performed without replacement is given by general multiplication rule as

So, here when second ball is drawn then it depend on the first ball because the for second ball only 4 balls are available in total as one red ball is already drawn put of which 1 ball is red and 3 balls are black.

(d)

Conditional probability means that the probability of an event is depend on the probability of the previous event happen.

In part (b), the probability P(B/A) is mothingnbut a conditional probability.

If happening of two events are not independent the their is need to use Conditional Probability.

Now, as in the above example, if we drawn a red ball then without replacement we draw a black a ball, then the probability of black ball is depend on the probability of red ball.So, here we will use Conditional Probability .So, Probability of drawing black ball after drawing first ball without replacement is

But if after drawing first red ball, with replacement we draw a black a ball, then the probability of black ball is not depend on the probability of red ball.So, here we will use Conditional Probability .So, Probability of drawing black ball after drawing first ball with replacement is just the probability of drawing a black ball .