Given events J and K: P(J) = 0.18 ; P(K) = 0.37 ; P(J or K) = 0.45
Find P(J and K)
Find the probability of the complement of event (J and K)
Find the probability of the complement of event (J or K)
P(J or K) = P(J) + P(K) − P(J and K); 0.45 = 0.18 + 0.37 − P(J and K) ; solve to find P(J and K) = 0.10
P( NOT (J and K) ) = 1 − P(J and K) = 1 − 0.10 = 0.90
P( NOT (J or K) ) = 1 − P(J or K) = 1 − 0.45 = 0.55
United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According to their website, http://www.unitedbloodservices.org/humanbloodtypes.html, a person with type O blood and a negative Rh factor (Rh−) can donate blood to any person with any bloodtype. Their data show that 43% of people have type O blood and 15% of people have Rh− factor; 52% of people have type O or Rh− factor.
Find the probability that a person has both type O blood and the Rh− factor
Find the probability that a person does NOT have both type O blood and the Rh− factor.
P(Type O or Rh−) = P(Type O) + P(Rh−) − P(Type O and Rh−)
0.52 = 0.43 + 0.15 − P(Type O and Rh−); solve to find P(Type O and Rh−) = 0.06
6% of people have type O Rh− blood
P( NOT (Type O and Rh−) ) = 1 − P(Type O and Rh−) = 1 − 0.06 = 0.94
94% of people do not have type O Rh− blood
At a college, 72% of courses have final exams and 46% of courses require research papers. Suppose that 32% of courses have a research paper and a final exam. Let F be the event that a course has a final exam.
Let R be the event that a course requires a research paper.
Find the probability that a course has a final exam or a research project.
Find the probability that a course has NEITHER of these two requirements.
P(R or F) = P(R) + P(F) − P(R and F) = 0.72 + 0.46 − 0.32 = 0.86
P( Neither R nor F ) = 1 − P(R or F) = 1 − 0.86 = 0.14
In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. Of those, 8% contain both chocolate and nuts. Sean is allergic to both chocolate and nuts.
Find the probability that a cookie contains chocolate or nuts (he can't eat it).
Find the probability that a cookie does not contain chocolate or nuts (he can eat it).
Let C be the event that the cookie contains chocolate. Let N be the event that the cookie contains nuts.
P(C or N) = P(C) + P(N) − P(C and N) = 0.36 + 0.12 − 0.08 = 0.40
P( neither chocolate nor nuts) = 1 − P(C or N) = 1 − 0.40 = 0.60
A college finds that 10% of students have taken a distance learning class and that 40% of students are part time students. Of the part time students, 20% have taken a distance learning class. Let D = event that a student takes a distance learning class and E = event that a student is a part time student
Find P(D and E)
Find P(E | D)
Find P(D or E)
Using an appropriate test, show whether D and E are independent.
Using an appropriate test, show whether D and E are mutually exclusive.
P(D and E) = P(D|E)P(E) = (0.20)(0.40) = 0.08
P(E|D) = P(D and E) / P(D) = 0.08/0.10 = 0.80
P(D or E) = P(D) + P(E) − P(D and E) = 0.10 + 0.40 − 0.08 = 0.42
Not Independent: P(D|E) = 0.20 which does not equal P(D) = .10
Not Mutually Exclusive: P(D and E) = 0.08 ; if they were mutually exclusive then we would need to have P(D and E) = 0, which is not true here.
When the Euro coin was introduced in 2002, two math professors had their statistics students test whether the Belgian 1 Euro coin was a fair coin. They spun the coin rather than tossing it, and it was found that out of 250 spins, 140 showed a head (event H) while 110 showed a tail (event T). Therefore, they claim that this is not a fair coin.
Based on the data above, find P(H) and P(T).
Use a tree to find the probabilities of each possible outcome for the experiment of tossing the coin twice.
Use the tree to find the probability of obtaining exactly one head in two tosses of the coin.
Use the tree to find the probability of obtaining at least one head.
P(H) = 140/250;
P(T) = 110/250
308/625
504/625
A box of cookies contains 3 chocolate and 7 butter cookies. Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it also. (How many cookies did he take?)
Draw the tree that represents the possibilities for the cookie selections. Write the probabilities along each branch of the tree.
Are the probabilities for the flavor of the SECOND cookie that Miguel selects independent of his first selection? Explain.
For each complete path through the tree, write the event it represents and find the probabilities.
Let S be the event that both cookies selected were the same flavor. Find P(S).
Let T be the event that both cookies selected were different flavors. Find P(T) by two different methods: by using the complement rule and by using the branches of the tree. Your answers should be the same with both methods.
Let U be the event that the second cookie selected is a butter cookie. Find P(U).
