Math 375 Exam 1 1. (10 points) A die is thrown 5 times. What is the probability
ID: 3305446 • Letter: M
Question
Math 375 Exam 1 1. (10 points) A die is thrown 5 times. What is the probability that numbers 1 and 6 occur at least once? 2. (10 points) You toss a coin three times. Given that you have obeerved at least one heads, what is the probability that you observe at least two heads? 3. (10 points) I roll a fair die twice and obtain two numbers: z result of the first roll and z result of the second roll Given that I know yy7 what is the probability that 3 or 37 4. (10 points) The weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time, when it does not rain, he incorrectly forecasts rain 10% of the time, what is the probability that it will rain tomorrow? 5. (10 points) A father of two children is picked at random. He then tells you, "I have two children, and one of them is a girl born on a Tuesday." Assuming he is telling the truth, what is the probability that his second child is also a girl?Explanation / Answer
4.
Solution: The sample space is defined by two mutually-exclusive events - it rains or it does not rain. Additionally, a third event occurs when the weatherman predicts rain. Notation for these events appears below.
In terms of probabilities, we know the following:
We want to know P( A1 | B ), the probability it will rain tomorrow, given a forecast for rain by the weatherman. The answer can be determined from Bayes' theorem, as shown below.
5. Sample space of a father having 2 childern is
G(Girl).B(Boy)
{GG,BB,GB,BG}
Now given that the first child is a girl we have 2 case {GG,GB}
P(second is a girl)= 1/2=0.5 ({GG}
P( A1 | B ) = P( A1 ) P( B | A1 )P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 ) P( A1 | B ) = (0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ] P( A1 | B ) = 0.111=11.1%