3.3 Conditional probability practice 2

Let [math]A[/math] and [math]B[/math] be events satisfying [math]P(A|B)=P(B|A)=0.4[/math] and [math]P(A\cup B)=1[/math]

(a) Find [math]P(A)[/math] and [math]P(B)[/math].

(b) Are [math]A[/math] and [math]B[/math] mutually exclusive? Write “Yes” or “No”.

20 students take AP calculus and AP statistics. 8 students got full scores in both AP calculus and AP statistics. 4 students got a full score in AP statistics but did not get a full score in AP calculus. 2 students got full scores in neither AP calculus nor AP statistics.

(a) Choose a student randomly. Given that the student got a full score in only one exam, find the probability that the student got a full score in AP statistics.

(b) Choose two students randomly. Find the probability that at least one student got full scores in both exams.

Ms. Cow has a machine that makes milk tea and lattes. Since Ms. Cow is old, the probability of her pressing the wrong button is 0.2. Well… the machine is also old, so the machine also has a certain chance of making mistakes. If Ms. Cow presses the “milk tea” button, there is a 0.4 chance that the machine makes latte instead; if she presses the “latte” button, there is a 0.3 chance that the machine makes milk tea instead. Given that Ms. Cow got milk tea, what is the probability that she pressed the “milk tea” button?

Kelly conducted a poll of 175 supermarkets has discovered that 115 of them sell [math]H\ddot{a}agenDazs[/math] ice cream and 122 of them sells [math]M\ddot{o}venpick[/math] ice cream.

(a) How many supermarkets sell both if each supermarket sells at least one of the two brand of ice cream?

(b) What is the probability that a store sells both brands given that they sell [math]M\ddot{o}venpick[/math]?

If events [math]A[/math] and [math]B[/math] are given such that [math]P(A)=x[/math], [math]P(A\cup B)=y[/math] and [math]P(B)=z[/math], explain how you can mathematically prove the following:

(a) Event [math]A[/math] and [math]B[/math] are independent.

(b) Event [math]A[/math] and [math]B[/math] are mutually exclusive.

Given the two events [math]A, B[/math] such that [math]P(A|B)=0.2[/math] and [math]P(B|A)=0.7[/math]. If events [math]A[/math] and [math]B[/math] are independent, find

(a) [math]P(A\cap B)[/math]

(b) [math]P(B)[/math]

(c) [math]P(B’\cap A’)[/math]. ([math]A'[/math] is the complement of [math]A[/math])

(Calculator) Given that [math]P(C)=\dfrac{9}{20}, P(C| D’)=\dfrac{3}{7} and P(C| D)=\dfrac{6}{13}[/math]

(a) Find P(D)

(b) Find [math]P(C’ \cap D’)[/math]

(c) Are [math]C[/math] and [math]D[/math] independent? Explain.

In a group of 40 students, 22 are in Honors Algebra 2, 25 take AP World History and 3 take neither of these subjects. Determine the probability that a randomly chosen student is in:

(a) Both Honors Algebra 2 and AP World History.

(b) At least one of these subjects.

(c) Honors Algebra 2 given that he or she is in AP World History.

(Calculator) Every Monday there is a [math]\dfrac{2}{5}[/math] probability that Alex is sleep deprived in class. There is only a [math]\dfrac{1}{10}[/math] chance that he is sleep deprived in class on any other weekday.

(a) What is the probability that Alex is sleep deprived in class?

(b) If Alex is sleep deprived, what is the probability it is a Monday?