1. Suppose a randomly selected passenger is about to go through the metal detector at Fuaʻamotu International Airport in Tonga. Consider the following two outcomes: The passenger sets off the metal detector, and the passenger does not set off the metal detector. Are these two outcomes equally likely? Explain why or why not. If you are to find the probability of these two outcomes, would you use the classical approach or the relative frequency approach or subjective approach? Explain why
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
1. Suppose a randomly selected passenger is about to go through the metal detector at Fuaʻamotu
International Airport in Tonga. Consider the following two outcomes: The passenger sets off the
metal detector, and the passenger does not set off the metal detector. Are these two outcomes
equally likely? Explain why or why not. If you are to find the probability of these two
outcomes, would you use the classical approach or the relative frequency approach or subjective
approach? Explain why
2.A company is to hire two new employees. They have prepared a final list of eight candidates, all
of whom are equally qualified. Of these eight candidates, five are women. Suppose the
company decides to select two persons randomly from these eight candidates.
a. What is the probability that:
i. Both candidate selected are women?
ii. At least one candidate selected is a woman?
iii. Second candidate is a woman.
iv. First candidate is a woman given that the second candidate is a woman
b. Let X denote the number of women in this sample.
i. Write the probability distribution of X.
ii. Find the standard deviation of X.
3.A trimotor plane has three engines—a central engine and an engine on each wing. The plane
will crash only if the central engine fails and at least one of the two wing engines fails. The 2 probability of failure during any given flight is .005 for the central engine and .008 for each of
the wing engines. Assuming that the three engines operate independently, what is theprobability that the plane will crash during a flight?
4. Many states in U.S.A have a lottery game, usually called a Pick-4, in which you pick a fourdigit number such as 7359. During the lottery drawing, there are four bins, each containing balls
numbered 0 through 9. One ball is drawn from each bin to form the four-digit winning number.
a. You purchase one ticket with one four-digit number. What is the probability that you will
win this lottery game? (2 marks)
b. There are many variations of this game. The primary variation allows you to win if the four
digits in your number are selected in any order as long as they are the same four digits as
obtained by the lottery agency. For example, if you pick four digits making the number
1265, then you will win if 1265, 2615, 5216, 6521, and so forth, are drawn. The variations
of the lottery game depend on how many unique digits are in your number. Consider the
following four different versions of this game. Find the probability that you will win this
lottery in each of these four situations.
i. All four digits are unique (e.g., 1234)
ii. Exactly one of the digits appears twice (e.g., 1223 or 9095)
iii. Two digits each appear twice (e.g., 2121 or 5588)
5. The weight of a sophisticated running shoe is
and a standard deviation of 0.5 ounces.
a. What is the probability that a shoe weighs more than 13 ounces?
b. What must the standard deviation of weight be in order for the company to state that 99.9%
of its shoes are less than 13 ounces?
6. A random sample of 20 acres gave a mean yield of sugarcane equal to 41.2 tons per acre with a
standard deviation of 3 tons. Assuming that the yield of sugarcane per acre is normally
distributed,
a. Construct a 90% confidence interval for the population mean.
b. What is the margin of error for this estimate in part a?
7. According to the Fiji Diabetes Association, 23.1% of Fijians aged 60 years or older had diabetes
in 2017. A recent random sample of 200 Fijians aged 60 years or older showed that 52 of them
have diabetes. Using a 5% significance level, perform a test of hypothesis to determine if the
current percentage of Fijians aged 60 years or older who have diabetes is higher than that in 2017. Use the P-value method.
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