Solve all Q 1 Seventy five percent of the defendants are, in fact, guilty Given that the defendant is guilty, there is a 70 percent chance the jury will convict the person Given that the defendant is not guilty, there is a 40 percent chance the jury will convict the person For simplicity, assume that the only options available to the jury are to convict or to release the defendant (a) What proportion of the defendants will be convicted by the jury (b) Given that a defendant is convicted, what is the probability the person is, in fact, guilty (c) What is the probability that the jury will make a correct decision (d) Given that the jury makes an incorrect decision, what is the probability that the decision is to release a guilty person 25 Bert computes a 95 confidence interval for p and obtains the interval 0 600, 0 700 Note Parts (a) and (b) are not connected Part (b) can be answered even if one does not know how to do part (a) (a) Bert's boss says, Give me a 90 confidence interval for p Calculate the answer for Bert (b) Bert's boss says, Give me a 95 confidence interval for p q Calculate the answer for Bert (Hint p q p (1 p) 2p 1 Bert's interval says, in part, that p is at least 0 600 what does this tell us about 2p 1 ) 26 Maggie computes a 95 confidence interval for p and obtains the interval 0 50, 0 75 Note Parts (a) and (b) are not connected Part (b) can be answered even if one does not know how to do part (a) (a) Maggie's boss says, Give me a 95 confidence interval for p 2 Calculate the answer for Maggie (Hint The interval says, in part, that p is at most 0 75 what does this tell us about p 2 ) (b) Maggie's boss says, Give me a 95 confidence interval for p q Calculate the answer for Maggie (Hint p q p (1 p) 2p 1 The interval says, in part, that p is at most 0 75 what does this tell us about 2p 1 ) 155 SECTION 3 2 Conditional Probability and the Multiplication Rule 27 Blood Types The probability that an African American person in the United States has type Of blood is 47 Six unrelated African American people in the United States are selected at random (Source American National Red Cross) (a) Find the probability that all six have type O blood (b) Find the probability that none of the six have type O blood (c) Find the probability that at least one of the six has type O blood inmal EvplainThe folowing concepts mathematically and give some examples (numerically) which are necessary in probability theory event sample sponce statistical inference descriptive s permutasyou kon binasyon clensity function conditional probability multiplication rule dependent event Independent event Boyes Rule P s The find and discuit the validated of Bayes' Rule 2 A bag contains 5 coins, one of which has a head on both sides while the other 4 coins are normal A coin is chosen at random from the bag and tossed 2 times The number of heads obtained is a random variable, say X (a) What are the possible values of X Also tabulate the pmf of X (Him The coin chosen is either normal or with head on both sides Find the conditional probability of X for each of these two situations Then use the law of total probability and multiplication rule to nd the probability of X ) (b) Calculate E(X) and Var(X)

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Solve all Q.1 Seventy-five percent of the defendants are, in fact, guilty. Given that the defendant is guilty, there is a 70 percent chance the

Solve all

Q.1 Seventy-five percent of the defendants are, in

fact, guilty. Given that the defendant is guilty,

there is a 70 percent chance the jury will convict the person. Given that the defendant is not

guilty, there is a 40 percent chance the jury will

convict the person.

For simplicity, assume that the only options

available to the jury are: to convict or to release

the defendant.

(a) What proportion of the defendants will be

convicted by the jury?

(b) Given that a defendant is convicted, what

is the probability the person is, in fact,

guilty?

make a correct decision?

(d) Given that the jury makes an incorrect decision, what is the probability that the decision is to release a guilty person?

25. Bert computes a 95% confidence interval for p

and obtains the interval [0.600, 0.700]. Note:

Parts (a) and (b) are not connected: Part (b) can

be answered even if one does not know how to

do part (a).

(a) Bert's boss says, "Give me a 90% confidence interval for p." Calculate the answer

for Bert.

(b) Bert's boss says, "Give me a 95% confidence interval for p?q." Calculate the answer for Bert. (Hint: p?q = p?(1?p) =

2p ? 1. Bert's interval says, in part, that

"p is at least 0.600;" what does this tell us

about 2p ? 1?)

26. Maggie computes a 95% confidence interval for

p and obtains the interval [0.50, 0.75]. Note:

Parts (a) and (b) are not connected: Part (b) can

be answered even if one does not know how to

do part (a).

(a) Maggie's boss says, "Give me a 95% confidence interval for p

." Calculate the answer for Maggie. (Hint: The interval says,

in part, that "p is at most 0.75;" what does

this tell us about p

(b) Maggie's boss says, "Give me a 95% confidence interval for p ? q." Calculate the

answer for Maggie. (Hint: p ? q = p ?

(1 ? p) = 2p ? 1. The interval says, in

part, that "p is at most 0.75;" what does

this tell us about 2p ? 1?)

155 SECTION 3.2 Conditional Probability and the Multiplication Rule 27. Blood Types The probability that an African American person in the United States has type Of blood is 47%. Six unrelated African American people in the United States are selected at random. (Source: American National Red Cross) (a) Find the probability that all six have type O+ blood. (b) Find the probability that none of the six have type O+ blood. (c) Find the probability that at least one of the six has type O+ blood. inmal" EvplainThe folowing concepts mathematically and give some examples (numerically) which are necessary in probability theory event sample sponce - statistical inference descriptive s. permutasyou kon binasyon - clensity function - conditional probability - multiplication rule - dependent event - Independent event Boyes Rule . P.s ; The find and discuit the validated of Bayes' Rule .2. A bag contains 5 coins, one of which has a head on both sides while the other 4 coins are normal. A coin is chosen at random from the bag and tossed 2 times. The number of heads obtained is a random variable, say X. (a) What are the possible values of X ? Also tabulate the pmf of X. (Him: The coin chosen is either normal or with head on both sides. Find the conditional probability of X for each of these two situations. Then use the law of total probability and multiplication rule to nd the probability of X.) (b) Calculate E(X) and Var(X)