Question

8. The Probability Calculus - Bayes's Theorem

Bayes's Theorem is used to calculate the conditional probability of two or more events that are mutually exclusive and jointly exhaustive. An event's conditional probability is the probability of the event happening given that another event has already occurred. The probability of event A given event B is expressed as P(A given B). If two events are mutually exclusive and jointly exhaustive, then one and only one of the two events must occur.

Your textbook restricts the number of mutually exclusive and jointly exhaustive events to two. When two events (A11 and A22) are mutually exclusive and jointly exhaustive, Bayes's Theorem is expressed as follows:

Consider the following scenario. Determine the fractional probabilities for the following events: P(A11), P(A22), P(B given A11), P(B given A22), as indicated next. Reduce the fractional answers to the lowest whole numbers and type your responses into the spaces provided. Then calculate the decimal probability of event P(A11 given B) using Bayes's Theorem. Type your numeric answer, written out to four decimal places, into the last space indicated.

Suppose you are given seven blue urns and three green urns, in a dark room so that you cannot tell the difference between the urns. Each blue urn contains one red ball and nine yellow balls. Each green urn contains four red balls and one yellow ball. You draw a ball at random from one of the urns.

Let event A11 be drawing a ball from a blue urn. What is the probability of event A11 expressed as a fraction? P(A11) = /

Let event A22 be drawing a ball from a green urn. What is the probability of event A22 expressed as a fraction? P(A22) = /

Let event B be drawing a red ball. Assuming the urn is blue, what is the probability of drawing a red ball from the blue urn, expressed as a fraction? P(B given A11) = /

Let event B be drawing a red ball. Assuming the urn is green, what is the probability of drawing a red ball from the green urn, expressed as a fraction? P(B given A22) = /

Assuming you draw a red ball from one of the urns, what is the (decimal) probability that the urn was blue according to Bayes's Theorem? P(A11 given B) =

Answer #1

There are 7 ways of selecting blue urns out of 10, therefore (A11) = 7/10

There are 3 ways of selecting green urns out of 10, therefore, P(A22) = 3/10

Since there is 1 way of selecting red ball from blue urn, P(B given A11) = (1/10)

Since there are 4 ways of selecting red ball from green urn, P(B given A22) = (4/5)

P(A11 given B) = P(B given A11)*P(A11) / P(B)

P(B) = P(B given A11).*P(A11) + P(B given A22)* P(A22)

= (1/10)*(7/10) + (4/5)*(3/10)

= 31/100

P(A11 given B) = (7/100) / (31/100) = 7/31

Use the negation rule, together with one or more of the other
rules of the probability calculus, to determine the probability
that the given event does occur. First indicate the probability
that the event does not occur. Then indicate the probability that
the event does occur by subtracting the first fraction from 1.
Reduce all fractions to the lowest whole numbers. Indicate your
answers by typing numeric responses in the spaces provided.
Consider a jar containing two blue balls, two...

An urn has n − 3 green balls and 3 red balls. Draw balls with
replacement. Let B denote the event that a red ball is seen at
least once. Find P(B) using the following methods
.(a) Use inclusion-exclusion with the events Ai = {ith draw is
red}. Hint. Use the general inclusion-exclusion formula from Fact
1.26 and the binomial theorem from Fact D.2.
(b) Decompose the event by considering the events of seeing a
red ball exactly k times,...

An urn contains five blue, six green and seven red balls. You
choose five balls at random from the urn, without replacement (so
you do not put a ball back in the urn after you pick it), what is
the probability that you chose at least one ball of each
color?(Hint: Consider the events: B, G, and R, denoting
respectively that there are no blue, no green and no red balls
chosen.)

You draw a card from an ordinary 52 card deck. Let P (J) be the
probability of drawing a jack. Let P K) be the probability of
drawing a king. Let P (H) be the probability of drawing a heart.
You draw one card. Answer each of the following based these
probabilities.
For a single trial, explain which events [P (J), P (K), P (H)]
are mutually exclusive and which events are not? Explain.
For a single draw, state the...

Consider an experiment where 2 balls are drawn from a bin
containing 3 red balls and 2 green balls (balls are not replaced
between drawn). Define the events A, B, and C as follows:
A = both balls are red, B = both balls are green, C = first ball
is red
A.) what is the sample space for this experiment (make a tree
diagram if needed)?
B.) What is the probability associated with each of the sample
points?
C.)...

Let A and B be two independent events such that P(A) = 0.17 and
P(B) = 0.63. What is P(A or B)? Your answer should be given to 4
decimal places.
An urn contains 27 red marbles, 20 blue marbles, and 42 yellow
marbles. One marble is to be chosen from the urn without
looking.
What is the probability of choosing a red or a blue marble?
Your answer should be rounded to 4 decimal places.

Urn A has 1 red and 2 black balls. Urn B has 2 red and 1 black
ball. It is common knowledge that nature chooses both urns with
equal probability, so P(A)=0.5 and P(B)=0.5. A sequence of six
balls is drawn with replacement from one of the urns. Experimental
subjects do not know which urn the balls are drawn from. Let x
denote the number of red balls that come up in the sample of 6
balls, x=0,1,2,...,6. Suppose that...

Assume you have a group of ten ping-pong balls numbered from 1-10.
Let us define some events as follows:
Event A: A
randomly selected ball has an odd number on it
Event B: A
randomly selected ball has a multiple of 3 on it
Event C: A randomly selected ball has either a 5 or 8 on it
Answer
the following questions related to probability of the above
described events for a SINGLE trial:
a.
P(A) =
(Remember
this is...

1. Suppose that A, B are two independent events, with
P(A) = 0.3 and P(B) = 0.4.
Find P(A and B)
a. 0.12
b. 0.3
c. 0.4
d. 0.70
2. Experiment: choosing a single ball from a bag which
has equal number of red, green, blue, and white ball and then
rolling a fair 6-sided die.
a.) List the sample space.
b.) What is the probability of drawing a green ball and
even number?
2.

1.
a.) Suppose you draw 8 cards from a standard deck of 52 cards,
one after the other, without replacement. Find the probability that
the last card is a club given that the first 7 cards are clubs.
b.) An urn contains 6 green, 10 blue, and 17 red balls. You take
3 balls out of the urn, one after the other, without replacement.
Find the probability that the third ball is green given that the
first two balls were...

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