(a) let {a1, a2} be a partition of a sample space and let b be any event. state and prove the law of total probability as it applies to the partition {a1, a2} and the event b.
there are two coins in a box; one coin is a fair coin, and the other is a biased coin thatcomes up heads 75 percent of the time. fiona chooses a coin at random and tosses it.
(b) what is the probability that the chosen coin flips heads? (hint: the law of totalprobability.)
(c) given that the chosen coin flips heads, what is the conditional probability that it wasthe biased coin? (hint: bayes’ theorem.)
(d) given that the chosen coin flips tails, what is the conditional probability that it wasthe biased coin?

:24% of students in a class are girls. if there are less than 30 students in this class, how many students are there? how many girls are there?

Listed below are time intervals (min) between eruptions of a geyser. assume that the "recent" times are within the past few years, the "past" times are from around 20 years ago, and that the two samples are independent simple random samples selected from normally distributed populations. do not assume that the population standard deviations are equal. does it appear that the mean time interval has changed? is the conclusion affected by whether the significance level is 0.10 or 0.01? recent 78 90 90 79 57 101 62 87 71 87 81 84 57 80 74 103 62 past 88 89 93 94 65 85 85 92 87 91 89 91 follow the steps of hypothesis testing, including identifying the alternative and null hypothesis, calculating the test statistic, finding the p-value, and making a conclusions about the null hypothesis and a final conclusion that addresses the original claim. use a significance level of 0.10. is the conclusion affected by whether the significance level is 0.10 or 0.01? answer choices below: a) yes, the conclusion is affected by the significance level because h0 is rejected when the significance level is 0.01 but is not rejected when the significance level is 0.10. b) no, the conclusion is not affected by the significance level because h0 is not rejected regardless of whether a significance level of 0.10 or 0.01 is used. c) yes, the conclusion is affected by the significance level because h0 is rejected when the significance level is 0.10 but is not rejected when the significance level is 0.01. d) no, the conclusion is not affected by the significance level because h0 is rejected regardless of whether a significance level of 0.10 or 0.01 is used.

Two consecutive negative integers have a product of 30. what are the integers?