Question 1 [12 + 4 =16 marks]
A. Let A and B be two events such that P( A) 0.6 , P(B) 0.4 and P( A B) 0.10.
Calculate P( A B).
Calculate P( A | B).
iii. Are events A and B independent? Justify your answer.
iv. Are events A and B mutually exclusive events? Justify your
answer.
(2 + 2 + 3 + 3 = 10 marks)
B. A box contains 20 DVDs, 4 of which are defective.
i. If two DVDs are selected at random (without replacement) from this box, what is the probability that both are defective?
ii. If two DVDs are selected at random (with replacement) from this box, what is the probability that both are defective?
(3 + 3 = 6 marks)
Question 2 [ 6 + 8 + 4 = 18 marks]
A. Four factories are to be randomly selected from a list of eleven factories for a full health and safety inspection
i. Calculate the number of ways that four factories can be
selected.
ii. Two of the factories have the same owner. Calculate the
probability that both of these
factories with the same owner will be included in the inspection.
(3 +3=6marks)
B. The random variable X has the following probability distribution:
X |
1 |
2 |
3 |
P(X) |
k |
3k |
6k |
i. Show that k. must equal 0.1
ii. Calculate the mean of X
iii. Calculate the standard deviation of X .
(2 +3+3=8marks)
C. Joe is playing a game of chance at the hibiscus festival, costing $1 for each game. In the game two fair dice are rolled and the sum of the numbers that turned up is found. If the sum is seven, then Joe wins $5. Otherwise loses his money. Joe play the game 15 times. Find his expected profit or lose.
(4 marks)
Question 3 [4 + 12 = 16 marks]
The basketball player has a 75% chance of a successful shot. The shots are assumed to be independent of each other. Find the probability that 3 out of the next 4 shots are successful.
(4 marks)
The weight of potatoes produced by one particular farm are found to be approximately normally distributed, with the mean weight of 147 g and a standard deviation of 23g.
i. What proportion of the potatoes produced by this farm will
weigh less than 160 g?
ii. Of all the potatoes produced on the farm, 5% are considered too
light. What is the
maximum weight that will be considered too light?
(3 + 5 = 8 marks)
C. The amounts of electricity bills for all households in a city have a skewed probability distribution with mean of $140 and a standard deviation of $30. Find the probability that the mean amount of electric bills for a random sample of 75 households selected from the city will be within $6 of the population mean.
(4 marks)
I need the answer as fast as possible please Question.4 [16 Marks] Let X equal the weight of the soap contained in a box. Assume that the distribution of X is N(u= 6.05, o2 = 0.0004). (a) Compute the probability P(X < 6.0171) (b) A random sample of nine (9) boxes of soap is selected from the production line. Let Y equal the number of boxes that weigh less than 6.0171. What is the distribution of Y? (e) Find the...
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1. Given below are marks obtained by 20 students out of 25 in Mathematics test. 21 23 19 17 12 15 15 17 17 19 23 23 21 21 25 25 21 19 19 19 Calculate: i. measures of central tendency. (9 marks) ii. range and semi inter-quartile range. (6 marks) iii. variance and determine the shape of skewness. (5 marks) (b) A factory produces components of which 1% is defective. The components are packed in boxes of 10. A...
Binomial Events Past records over a long period of time show that 10.0% of the screws manufactured by a particular machine are defective. Given that a random sample of 10 screws are selected, complete the following tasks. a). Calculate the mean. b). Calculate the variance and standard deviation. c). Calculate the probability of 0 to 10 screws being defective [i.e. P(X=0), P(X=1), P(X=2)... P(X=n)]. d). Construct a histogram of the probability of faulty screws versus the number of faulty screws....
Problem #3: Let A and B be two events on the sample space S. Then show that a. P(B) P(AOB)+P(AnB) b. If Bc A, then show that P(A)2 P(B) Show that P(A| B)=1-P(A|B) C. P(A) d. If A and B are mutually exclusive events then show that P(A| AUB) = PA)+P(B) Problem 4: If A and B are independent events then show that A and B are independent. If A and B are independent then show that A and B...
ustion 2: (Discrete Random variable)[2+2-4 marks] A factory produces components of which I % is defective. The components are in boxes of 10 A box is selected at random (a) Find the probability that the box contains at least 2 defective components. (b) Find the mean and the standard deviation of the distribution Cy e length of life of an instrument produced by a machine has a normal distribution with a mean of 12 months and standard deviation of 2...
Q1) Consider two events P and Q. a. Write the general formula used to calculate the probability that either event P occurs or Q occurs or both occur. b. How does this formula change if: i. Events P and Q are disjoint (i.e., mutually exclusive of each other). ii. Events P and Q are nondisjoint events that are statistically independent of each other. iii. Events P and Q are nondisjoint events that are statistically dependent of each other. Q2) Rewrite...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R be a (differentiable) function and let Y = g(X). Write expressions for the following ((ii)-(iv) should be in terms of the density of X (i) The integral f()d (ii) The mean E(X) (ii The probability P(X e (a, b) (iv) The mean E(g(X)) R be a smooth (1 mark (1 mark) (1 mark (1 mark) (b) Let z E R be a constant and...
sorry R is 8 Question 5 10 marks (a) . Suppose X is a normally distributed random variable with a mean of 15 and a standard deviation of 2.34+G Find the following probabilities. Correct your answers to 4 decimal places. (1) (ii) P(X > 14.5) P(15.3 < X < 16.8) (2 marks) (3 marks) (b) Suppose X is the sample mean of 10 observations selected randomly from a normal distribution with a mean of 15 and a standard deviation of...
tnis p your answer sheets 1. A single card is drawn from 52-card deck (10) Let A denotes the event that the card is red and B denotes the e is spade. Are the events being? a) Mutually exclusive? b) Dependent? What is your conclusion and B? Any exception? 2. Two team (A and B) play a series of baseball games. The team who v five-game-series wins the series. Consider A has home-field advantag probability of winning 0.7 if it...