A standard deck of 52 playing cards consists of four suits:
♥, ♦,
♠, ♣.
Each suit includes thirteen cards. One of the cards in each suit is called the ace.
Billy picks one random card out of the standard deck of 52 playing cards.
(a)
Find the probability that this card is a heart: ♥.
(b)
Find the probability that this card is an ace.
Mandy picks two random cards out of the standard deck of 52 playing cards (without replacing).
(c)
Find the probability that these cards are two hearts.
(d)
Find the probability that at least one of these two cards is an ace.
Candace and Jeremy are investigating the reading habits in their town. The town has 8000 inhabitants
and there are two libraries: American Library and British Library.
American Library has 2300 members, British Library has 1900 members. There are 600 inhabitants
who are members of both libraries.
Candace selects a random citizen of this town.
(a)
Find the probability that this citizen is a member of exactly one library.
(b)
Find the probability that this citizen is a member of at least one library.
(c)
Find the probability that this citizen is not member of any library.
Jeremy selects a random member of the American Library.
(d)
Find the probability that this person is a member of the British Library, too.
Among 1200 students of the Superhero High School
840 attend the IB program and 360 attend the national program.
In IB program, 65% of students are female.
In national program, 55% of students are female.
(a)
We choose a random student of this school. Find the probability that:
(i)this student is female,
(ii)this student is male.
(b)
We choose a random student of this school. Find the probability that:
(i)this student attends the IB program,
(ii)this student attends the national program.
(c)
We choose a random female student of this school. Find the probability that:
Candidates for admission to the Tri-state University send their applications to the
admissions desk. There, randomly chosen 70% of the applications
are processed by Mr Goodfellow and others are processed by Mr Hellboy.
Mr Goodfellow accepts 90% of his applicants and Mr Hellboy accepts only 15% of his applicants.
All other applicants are rejected.
We can use a tree diagram to illustrate the situation:
(a)
Complete the tree diagram by writing down the corresponding probabilities.
Phineas submitted his application to the admissions desk.
(b)
Find the probability that he will be accepted.
Ferb submitted his application to the admissions desk, too. After a while he received the answer
that he was not accepted to the university.
(c)
Find the probability that his application was processed by Mr Hellboy.
Johnny Bravo started training basketball, but he is not good at it: he hits the basket in 70% attempts.
If he wants to join a basketball team called Kids Next Door, he must pass a test, where he will have 10 attempts.
If he hits the basket at least 8 times in 10 attempts, he will join the team.
(a)
Find his expected number of baskets in 10 attempts.
(b)
Find the probability that he will hit the basket in all 10 attempts.
(c)
Find the probability that he will pass the test (hit the basket for 8 or more times in 10 attempts).
Dexter is conducting an experiment in his laboratory.
He wants to explore how exposure to sun light affects the growth of tomato plants.
For each plant, he fixed a specific number of daily hours when the plant is exposed to sun light.
After a week he measured the growth of each plant in cm. Here are his results:
plant
A
B
C
D
E
daily exposure
4
5
6
8
10
growth (cm)
12
13
18
19
20
(a)(i)Find the Pearson product-moment correlation coefficient.
(ii)Explain the meaning of your result.
(b)
Draw the scatter plot.
(c)
Write the equation of the line of best fit.
(d)
Hence, predict the growth of a plant which would have 7 hours of daily exposure to sun light.
Laundry detergent factory Doofenschmirtz Evil Inc. produces washing powder in 5 kg packages. But, packaging machines
cannot determine the exact quantity of 5 kg. The quantity of washing powder in a package is in fact
normally distributed and packaging machines are programmed so that the average is \(\mu=5.2~\mathrm{kg}\)
and standard deviation is \(\sigma=0.09~\mathrm{kg}\). In one day they produce 3500 packages.
(a)According to EU standards all packages containing less then 5 kg are discarded.
(i)Find the percent of discarded packages.
(ii)Find the number of packages discarded in one day.
Besides, the factory decided for a special treatment of the heaviest packages. The heaviest 5% of packages will get a special
sticker "Extra Large Pack" and will be sold as a special offer.
(b)Find the minimal weight of an Extra Large Pack.
Solutions:
(a) \(P(X\lt 5~\mathrm{kg})\approx1.31\%\) and it means 46 discarded packages daily;
(b) minimal weight is \(5.35~\mathrm{kg}\)
Students constructed a wheel of fortune with numbers 1, 2, 3, 4, 5.
Professor Utonium wanted to verify whether this wheel of fortune is fair or not. He spun the wheel 15 times and
recorded the results:
result
1
2
3
4
5
frequency
2
2
2
2
7
(a)Which statistical test can be used to determine whether the wheel of fortune is fair?
(b)Write down the null hypothesis.
(c)Write down the degree of freedom.
(d)Calculate the p-value and explain if the null hypothesis should be rejected or not.
Use the standard level of significance.
Professor Utonium decided to continue his research. He continued spinning the wheel until he reached a total of 60 spins.
Here are the results he recorded:
result
1
2
3
4
5
frequency
9
10
9
10
22
(e)Calculate the p-value for this set of data
and explain if the null hypothesis should be rejected now.
Solutions:
(a) \(\chi^2\)-GOF test;
(b) \(H_0:\) This wheel of fortune is fair;
(c) \(df=4\);
(d) \(pVal\approx15.5\%\). We can't reject the null hypothesis, because p-value
is not less than the standard level of significance (5%).
(e) \(pVal\approx3.28\%\lt5\%\). Now we have enough data to reject the null hypothesis. This wheel of fortune is not fair.
