A box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places.
The probability is \(P(R)+P(B)\). This is \(\frac{54}{138}+\frac{75}{138}=\frac{129}{138}\)
You are going to play mini golf. A ball machine that contains 19 green golf balls, 20 red golf balls, 24 blue golf balls, and 17 yellow golf balls, randomly gives you your ball. What is the probability that you end up with a red golf ball? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
The probability is \(\frac{20}{71}\)
A pizza delivery company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 1399 customers. What is the probability that a customer is not male or does not live with parents? Write your answer as a fraction or a decimal number rounded to four decimal places.
Let \(P(M)\) be the probability of a male customer and \(P(WP)\) be the probability of the customer living with parents.
For some reason, Rstudio is not formatting this piece of LaTeX properly. I have added the raw LaTeX in a code box, so it can be seen. The answer is \(\frac{536}{1399}\)
\[
P(M^\prime)+P(WP^\prime)=1-(P(M)+P(WP)-P(M \cup WP))\\
P(M)=\frac{671}{1399}\\
P(WP)= \frac{467}{1399}\\
P(M\cup WP)=\frac{215}{1399}\\
1-(P(M)+P(WP)-P(M \cup WP))=\\
1-\frac{671}{1399}-\frac{467}{1399}+\frac{215}{1399}=\\
\frac{536}{1399}
\] [ P(M)+P(WP)=1-(P(M)+P(WP)-P(M WP))\
P(M)=\
P(WP)= \
P(MWP)=\
1-(P(M)+P(WP)-P(M WP))=\
1--+=\
]
Determine if the following events are independent.
Going to the gym. Losing weight.
These are dependent. Even if the exercise is not directly contributing to weight loss, it will change the composition of the body and change the difficulty in shedding weight.
A veggie wrap at City Subs is composed of 3 different vegetables and 3 different condiments wrapped up in a tortilla. If there are 8 vegetables, 7 condiments, and 3 types of tortilla available, how many different veggie wraps can be made?
\(\binom{8}{3} \times\binom{7}{3}\times\binom{3}{1}=56\times 35 \times 3=5880\)
Determine if the following events are independent.
Jeff runs out of gas on the way to work. Liz watches the evening news.
There is insufficient information to decide if the two are independent. Jeff could work a swing shift and running out of gas caused a traffic jam large enough that Liz noticed it, was curious about it, and it was displayed on the evening news. Or Jeff could have run out of gas because of a shortage and Liz watches the news to find out about the shortage. The two are probably independent, but there are scenarios by which they can be connected.
The newly elected president needs to decide the remaining 8 spots available in the cabinet he/she is appointing. If there are 14 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?
\(P(14,8)=\frac{14!}{(14-8)!}=14\times 13\times 12 \times 11 \times 10 \times 9 \times 8 \times 7= 121080960\)
A bag contains 9 red, 4 orange, and 9 green jellybeans. What is the probability of reaching into the bag and randomly withdrawing 4 jellybeans such that the number of red ones is 0, the number of orange ones is 1, and the number of green ones is 3? Write your answer as a fraction or a decimal number rounded to four decimal places.
There are \(\binom{22}{4}\) combinations. There are \(\binom{9}{0}\times\binom{4}{1}\times\binom{9}{3}\) possible combinations. The probability is \(\frac{336}{7315}\).
Evaluate the following expression. \(\frac{11!}{7!}\)
\(11 \times 10 \times 9 \times 8 =7920\)
Describe the complement of the given event. 67% of subscribers to a fitness magazine are over the age of 34.
33% of subscribers to a fitness magazine are not over the age of 34. Alternatively, 33% of subscribers are 34 or younger.
If you throw exactly three heads in four tosses of a coin you win $97. If not, you pay me $30.
Step 1. Find the expected value of the proposition. Round your answer to two decimal places.
\(E(X)=97\times\frac{\binom{3}{4}}{2^4}=97\times\frac{4}{16}=97\times\frac{1}{4}=24.25\)
\(E(Y)=-30*\frac{3}{4}=22.5\)
Expected value is $24.25-$22.5 = $1.75
Step 2. If you played this game 559 times how much would you expect to win or lose? (Losses must be entered as negative.)
$1.75x559=$1048.25
Flip a coin 9 times. If you get 4 tails or less, I will pay you $23. Otherwise you pay me $26.
Step 1. Find the expected value of the proposition. Round your answer to two decimal places.
\[ E(X)=23\times\frac{\sum_{i=0}^4\binom{9}{i}}{2^9}=23\times\frac{1}{2}=11.5 \] \(E(Y)=-26*\frac{1}{2}=-13\)
Expected value is -$1.50
Step 2. If you played this game 994 times how much would you expect to win or lose? (Losses must be entered as negative.)
-$1.50x994=-$1491
The sensitivity and specificity of the polygraph has been a subject of study and debate for years. A 2001 study of the use of polygraph for screening purposes suggested that the probability of detecting a liar was .59 (sensitivity) and that the probability of detecting a “truth teller” was .90 (specificity). We estimate that about 20% of individuals selected for the screening polygraph will lie.
Let \(P(P)\) be the probability that the polygraph indicates lying, let \(P(N)\) be the probability that it indicates truth telling, let \(P(T)\) be the probability of telling the truth, and let \(P(L)\) be the probability of lying.
\[ P(L|P)=\frac{P(L)P(P|L)}{P(P)}\\ =\frac{.2\times .59}{P(L)P(L|P)+P(T)P(T|P)}\\ =\frac{.2\times .59}{.2\times .59 + .8 \times .1}\\ =\frac{.118}{.118+.08}\\ =59.6\% \]
\[P(T|N)=\frac{P(T)P(N|T)}{P(N)}\\ =\frac{P(T)P(N|T)}{P(L)P(N|L)+P(T)P(N|T)}\\ =\frac{.8\times.9}{.2\times.41+.8\times.9}\\ =89.8\% \]
\[ P(L\cup P)=P(L)+P(P)-P(L\cap P)\\ =.2+(.2\times.59+.8\times.1)-.118\\ =28\% \]