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Toss two fair coins and let X equal the number of heads observed .Find the probability distribution for X.

Let A and B be two events. Suppose that the probability that neither event occurs is3/8. What is the probability that at least one of the events occurs?

Suppose that one box contains 60 long bolts and 40 short bolts ,and that the other box contains 10 long bolts and 20 short bolts. Suppose also that one box is selected at random and a bolt is then selected at random from that box. Find the probability that
this bolt is long.

A boy goes to his school either by bus or on foot.If one day he goes to the school by bus,then the probability that he goes by bus the next day is 7/10.If one day he walks to school,then the probability that he goes by bus the next day is 2/5. given that he walks to the school on particular Tuesday,find the probability that he will go to the school by bus on Thursday of the week

Let X have MGF MX (t) =1/8(1+ e^t )
a) Find E(X) using MX(t)
b) Find the variance of X using MX(t).
c) Find the moment generating function of Y=2X-1.

Whilst shopping, the probability that Caroline buys fruit is 0.6.
The probability that Caroline buys a CD is 0.2.
Buying fruit and buying a CD are independent of each other.
Work out the probability that she buys fruit, a CD or both

A safety officer wants to prove that the average speed of cars driven by a school is lessthan 25 mph. Suppose that a random sample of 14 cars shows an average speed of 24.0 mph, with a sample standard deviation of 2.2 mph. Assume that the speeds of cars are normally distributed. What are the appropriate null and alternative hypotheses? Testing at significance level of α=5% if the p-value = 0.057, what is an appropriate conclusion?

Past data indicates an 80% (0.8) success rate in treating a certain medical problem. Researchers want to test if a new treatment will increase the success rate above 80%. This new treatment was used on 100 patients in a clinical trial and was successful on 87 of those patients. Write the null and alternative hypothesis and test at 5% significance level.

6. The time it takes for Norman to travel from his house to the 10:00 AM MATH 1F92 lecture is
approximately normal with a mean of 25 minutes and standard deviation of 3 minutes.
(Note: Labelled diagrams and proper notation are required for all parts.)
a) One morning, Norman leaves his house at 9:40 AM. The professor is going to give hints about
the test during the first 7 minutes of lecture. What is the probability that he will miss all the hints?
b) On the morning of the test, Norman wants to arrive by 9:30 AM. What time must Norman leave
his home to ensure a 97% probability that he will arrive before 9:30 AM?

The time it takes for Norman to travel from his house to the 10:00 AM MATH 1F92 lecture is
approximately normal with a mean of 25 minutes and standard deviation of 3 minutes.
(Note: Labelled diagrams and proper notation are required for all parts.)
a) One morning, Norman leaves his house at 9:40 AM. The professor is going to give hints about
the test during the first 7 minutes of lecture. What is the probability that he will miss all the hints?
b) On the morning of the test, Norman wants to arrive by 9:30 AM. What time must Norman leave
his home to ensure a 97% probability that he will arrive before 9:30 AM?