Question

Consider tossing a fair die two times. Let A = 3 or less on the first roll, B = sum of the two rolls is at least 10. Events A and B . Events A and B .

(a) are disjoint, are independent

(b) are disjoint, are not independent

(c) are not disjoint, are independent

(d) are not disjoint, are not independent

Consider tossing a fair die two times. Let A = 4 or less on the first roll, B = the first roll is the same as the second roll. Events A and B . Events A and B .

(a) are disjoint, are independent

(b) are disjoint, are not independent

(c) are not disjoint, are independent

(d) are not disjoint, are not independent

The probability that a randomly selected American has disease A is p = 0.15. If 15 people are randomly selected from the population, what is the probability that at most 13 of them do not have disease A?

You collect 300 votes: 100 votes from Pawnee citizens and 200 votes from Eagleton citizens. The table below provides the proportions of the Pawnee votes that were for Voldemort or Harry, as well as the proportion of the Eagleton votes that were for Voldemort or Harry. Find a value of p such that a (randomly selected) vote for Voldemort has a 0.9 probability of being an Eagletonian vote:

	Harry (H)	Voldemort (V)
Pawnee (P)	0.8	0.2
Eagleton (E)	1-p	p

You have become convinced that the weather depends on whether you bring an umbrella. To investigate, you keep track of when it rains and when you bring an umbrella. According to your records, it rained on 30% of days. On days when it rained, you had your umbrella with you 25% of the time. On days when it didn’t rain, you had your umbrella with you 50% of the time. You brought your umbrella today. What’s the probability that it rains?

Answer 1

1)

outcome of A = three or less on first roll (18 outcomes)

outcome of B={(4,6)(6,4)(5,5)(5,6)(6,5)(6,6)

there is no common outcome

so, answer is option b)
disjoint , are not independent
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2)

outcome of A = four or less on first roll (24 outcomes)

outcome of B = 6 outcomes

common outcomes = {(1,1)(2,2)(3,3)(4,4)}

there is common outcome

so, these are not disjoint

also,

P(A)P(B) = 24/36*6/36 = 0.111

P(A n B) = 4/36 = 0.111

since, P(A)P(B) = P(A n B)

so, these are independent events

so, answer is are disjoint, are independent