MATH 106 6383 Finite Mathematics (2192)
Week 5 Discussion
PROBABILITY (Applied Finite Mathematics, “Probability”)
A card is selected from a deck. Find the following probabilities.
P(a red card)
P(a face card)
P(a jack and a spade)
A jar contains 6 red, 7 white, and 7 blue marbles. If a marble is chosen at random, find the following probabilities.
P(red)
P(white)
P(red or blue)
P(red and blue)
Consider a family of three children. Find the following probabilities.
P(two boys and a girl)
P(at least one boy)
P(children of both sexes)
P(at most one girl)
Two dice are rolled. Find the following probabilities.
P(the sum of the dice is 5)
P(the sum of the dice is 8)
P(the sum is 3 or 6)
P(the sum is more than 10)
A jar contains four marbles numbered 1, 2, 3, and 4. If two marbles are drawn, find the following probabilities.
P(the sum of the number is 5)
P(the sum of the numbers is odd)
P(the sum of the numbers is 9)
P(one of the numbers is 3)
MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE
Determine whether the following pair of events are mutually exclusive.
A={A person earns more than $25,000}B={A person earns less than $20,000}
A card is drawn from a deck: C={It is a King}D={It is a heart}
A single, fair, 6-sided die is rolled: E= {An even number shows} F={A number greater than 3 shows}
Two fair 6-sided dice are rolled: G={The sum of dice is 8} H={One die shows a 6}
Three fair coins are tossed: I= {Two heads come up} J={At least one tail comes up}
A family has three children: K= {First born is a boy} L={The family has children of both sexes}
Use the addition rule to find the following probabilities.
A card is drawn from a 52-card deck, and the events C and D are as follows: C= {It is a king} D={It is a heart}. Find “the probability of event C OR event D occurring”, P(C U D).
A single fair 6-sided die is rolled, and the events E and F are as follows: E={An even number shows} F={A number greater than 3 shows}. Find “the probability of event E OR event F occurring”, P(E U F).
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
This quarter, there is a 50% chance that Jason will pass Accounting, a 60% chance that he will pass English, and 80% chance that he will pass at least one of these two courses. What is the probability that he will pass both Accounting and English?
The following table shows the distribution of Democratic and Republican U.S. Senators by gender.
MALES(M)
FEMALES(F)
TOTAL
DEMOCRATS(D)
39
4
43
REPUBLICANS(R)
52
5
57
TOTALS
91
9
100
Use the table to determine the following probabilities:
P(M ∩ D)
P(F ∩ R)
P(M U D)
P(M U F)
Use the addition rule to determine the following probabilities.
If P(E)= 0.5 and P(F)= 0.4 and E and F are mutually exclusive, find P(E ∩ F).
If P(E)= 0.4 and P(F)= 0.2 and E and F are mutually exclusive, find P(E U F).
Use the addition rule to determine the following probabilities.
If P(E)= 0.3 and P(E U F)= 0.6 and P(E ∩ F)= 0.2, find P(F).
If P(E)= 0.4, P(F)= 0.5 and P(E U F)= 0.7, find P(E and F).
CALCULATING PROBABILITIES USING TREE DIAGRAMS AND COMBINATIONS
A basket contains six red and four blue marbles. Three marbles are drawn at random. Find the following probabilities using the tree diagram. Do not use combinations.
P(All three red)
P(two red, one blue)
P(one red, two blue)
P(first red, second blue, third red)
A committee of four is selected from a total of 4 freshmen, 5 sophomores, and 6 juniors. Find the probabilities for the following events.
At least three freshmen.
All four of the same class.
Not all four from the same class.
Exactly three of the same class.
Five cards are drawn from a deck. Find the probabilities for the following events.
Two hearts, two spades, and one club.
A flush of any suit(all cards of a single suit).
A full house of nines and tens(3 nines and 2 tens).
A pair of nines and tens.
CONDITIONAL PROBABILITY
Do the following problems using the conditional probability formula:
PAB=PA∩BP(B) PBA=PA∩BP(A)
A single card is drawn from a 52-card deck. Find the conditional probability of P(a queen?a club).
If P(A)=.3 and P(B)=.4, and P(A and B)=.12, find the following.
P(A?B)
P(B?A)
At De Anza College, 20% of the students take Finite Math, 30% take History, and 5% take both Finite Math and History. If a student is chosen at random, find the following conditional probabilities.
Student is taking Finite Math given that said student is taking History.
Srudent is taking History given that said student is taking Finite Math.
At a college, 60% of the students pass Accounting, 70% pass English, and 30% pass both of these courses. If a student is selected at random, find the following conditional probabilities.
The student passes Accounting given that said student passed English.
The student passes English given that said student passed Accounting.
The following table shows the distribution of Democratic and Republican U.S. Senators by gender.
MALE(M)
FEMALE(F)
TOTAL
DEMOCRATS(D)
39
4
43
REPUBLICANS(R)
52
5
57
TOTALS
91
9
100
Use the table to determine the following probabilities:
P(M?D)
P(D?M)
P(F?R)
P(R?F)
If P(E)= 0.3, and P(F)= 0.3, and E and F are mutually exclusive, find P(E?F).
If P(E ∩ F)= 0.04 and P(E?F)= 0.1, find P(F).
INDEPENDENT EVENTS
For a two-child family, let the events E, F, and G be as follows.
E: The family has at least one boy
F: The family has children of both sexes
G: The family’s first born is a boy
Find the following.
P(E)
P(F)
P(E∩F)
Are E and F independent?
For a two-child family, let the events E, F, and G be as follows.
E: The family has at least one boy
F: The family has children of both sexes
G: The family’s first born is a boy
Find the following.
P(F)
P(G)
P(F∩G)
Are F and G independent?
If P(E)= 0.9, P(F?E)= 0.36, and E and F are independent, find P(F).
If P(E)= 0.6, P(E U F)= 0.8, and E and F are independent, find P(F).
In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 6 had minor headaches. Of the non-drinkers, 9 had minor headaches. Are the events “drinkers” and “had headaches” independent?
John’s probability of passing statistics is 40%, and Linda’s probability of passing the same course is 70%. If the two events are independent, find the following probabilities.
P(both of them will pass statistics)
P(at least one of them will pass statistics)
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EXPECTED VALUE(Applied Finite Mathematics, “More Probability”)
Do the following problems using the expected value concepts learned in this section
In a town, 40% of the men and 30% of the women are overweight. If the town has 46% men and 54% women, what percent of the people are overweight?
A game involves rolling a single die. One receives the face value of the die in dollars. How much should one be willing to pay to roll the die to make the game fair?
In a European country, 20% of the families have three children, 40% have two children, 30% have one child, and 10% have no children. On average, how many children are there to a family?
A game involves drawing a single card from a standard deck. One receives 60 cents for an ace, 30 cents for a king, and 5 cents for a red card that is neither an ace nor a king. If the cost of each draw is 10 cents, should one play? Explain.
During her four years at college, Niki received A’s in 30% of her courses, B’s in 60% of her courses, and C’s in the remaining 10%. If A=4, B=3, and C=2, find her grade point average.
A Texas oil drilling company has determined that it costs $25,000 to sink a test well. If oil is hit, the revenue for the company will be $500,000. If natural gas is found, the revenue will be $150,000. If the probability of hitting oil is 3% and of hitting gas is 6%, find the expected value of sinking a test well.
A $1 lottery ticket offers a grand prize of $10,000; 10 runner-up prizes each paying $1000; 100 third-place prizes each paying $100; and 1,000 fourth-place prizes each paying $10. Find the expected value of entering this contest if 1 million tickets are sold.
Assume that for the next heavyweight fight the odds of Mike Tyson winning are 15 to 2. A gambler bets $10 that Mike Tyson will lose. If Mike Tyson loses, how much can the gambler hope to receive?
PROBABILITY USING TREE DIAGRAMS
Use a tree diagram to solve the following problems.
A coin is tossed until a head appears. What is the probability that a head will appear in at most three tries?
John’s car is in the garage, and he has to take a bus to get to school. He needs to make all three connections on time to get to his class. If the chance of making the first connection on time is 80%, the second 80%, and the third 70%, what is the chance that John will make it to his class on time?
For a real estate exam the probability of a person passing the test on the first try is .70. The probability that a person who fails on the first try will pass on each of the successive attempts is .80. What is the probability that a person passes the test in at most three attempts?
The Long Life Light Bulbs claims that the probability that a light bulb will go out when first used is 15%, but if it does not go out on the first use the probability that it will last the first year is 95%, and if it lasts the first year, there is a 90% probability that it will last two years. What is the probability that a new bulb will last two years?
A die is rolled until an ace (1) shows. What is the probability that an ace will show on the fourth try?
