Download Binomial Distribution Tutorial 1

Binomial Distribution Tutorial 1
1. [J77/I/11]
A lake containes 1000 fish of species A and 2000 fish of species B and no other fish. Ten fish are
taken from taken from the lake at random.
(a) Explain why the probabilities of obtaining various numbers of fish of species A in the ten can
be very well approximated by a binomial probabilty distribution.
(b) Estimate to three significant figures the probability of obtaining 4 fish of species A and 6 of
species B.
(c) Estimate also to three significant figures the probability that in a catch of 5 fish altogether
there are more fish of species A than there are of species B.
[b) 0.228 c) 0.210]
2. [2012/RI/Prelim/II/11]
As part of a promotion, a large number of scratch-and-win cards are printed and one card is given
out with every value meal purchase in a restaurant. A restaurant sells 20 value meals per day and
the proportion of scratch-and-win cards with prizes is 0.1
(a) Show that the probability that more than 2 prizes are given out in 2 days is 0.777.
[2]
(b) 10 restaurants took part in the promotion and each sold 20 value meals per day.
Find the probability that at most 2 restaurants give out fewer than 3 prizes in 2 days.
[2]
[(b) 0.608 (3 s.f.)]
3. [07/TPJC/Prelim/II/5modified]
A factory produces chocolate which are packed into boxes of n chocolates and delivered to shops
for sale. A chocolate will not meet the minimum criteria for packing for sale if it weights less than
20 grams. On average, 2% of the chocolate produced did not meet the minimum criteria.
(a) Given that the probability that there are more than 2 chocolates in a box that do not meet
minimum criteria is less than 0.03, find the largest number of n.
[3]
(b) Given that the chocolates are packed into boxes of 20,
i. find the probability that a randonly chosen box contain at least 1 chocolate that does
not meet the minimum criteria,
[2]
ii. find the probability that out of 4 randomly chosen boxes of chocolate, there are exactly
2 boxes with at least 1 chocolate that does not meet the minimum criteria.
[2]
[a) 33 b)i) 0.332 ii) 0.295]
4. [2012/SAJC/Prelim/II/13]
On the average, 30% of the students in Calculus Madness Institute (CMI) could do the Differential
Equation question in Block Test Two. The principal randomly select a class of 30 students to
analyse the results.
(a) State, in the context of this question, two assumptions needed to model the results by a
binomial distribution.
[2]
(b) Find the probability that at least 6 students in that class could do that question.
[2]
(c) *Find the probability that only 2 students among the first 8 selected students in that class
could do the question given that at least 6 students could do that question.
[3]
(d) The probability of no more than 5 students could do the question in a randomly selected
class exceeds 0.9. Find the largest possible number of students in that class.
[3]
[(b) 0.923 (c) 0.299 (d) 11]
c 2017 Math Academy
www.MathAcademy.sg
1
www.MathAcademy.sg
5. [2008/MJC/II/Q8]
In a school with a large number of students, on average, 1 out of 5 uses AIKON brand mobile
phone. In a random sample of n students, find the least value of n such that the probability that
at least one student uses AIKON brand mobile phone exceeds 0.99.
[least n = 21 ]
6. [2011/JJC/Prelim/II/9]
A large batch of beans contains only red beans and green beans. The proportion of red beans is
p. A sample of ten beans is taken from this batch at random.
(a) Explain why the probabilities of obtaining various numbers of red beans in the sample can
be well approximated by a binomial probability distribution.
[1]
(b) Given that the probability of obtaining at most one red bean in the sample is 0.96, show that
p satisfies the equation
25(1 − p)9 (1 + 9p) − 24 = 0
and hence find the value of p.
[4]
[(b) p = 0.0325]
7. [2012/IJC/Prelim/II/7]
(a) A die is biased and the probability, p, of throwing a six is known to be less than 61 . An
experiment consists of recording the number of sixes in 25 throws of the die. In a large
number of experiments, the standard deviation of the number of sixes is 1.5. Show that the
1
. Hence find the probability that at least 6 but fewer than 10 sixes are reocrded
value of p is 10
during a particcuar experiment.
[4]
(b) The biased die is now thrown 40 times. Find the most likely number of sixes obtained.
[1]
Note: Variance = (Standard deviation)2
[(a) 0.0333 (b) 4]
8. [2011/DHS/Prelim/II/5]
In a binomial probability distribution X, there are n trials and the probability of success of each
trial is p. If n = 20 and P(X ≤ 1) = 0.8, determine the value of p. Hence find the least value of a
such that P(X < a) > 0.999.
[5]
[Ans: 6]
9. [2009/NYJC/II/9b]
At the onset of a particular contagious flu virus, the probability that a patients survives is given
by p. In a hospital ward of 50 patients, past data shows that the probability of more than 48
patients surviving the flu is 0.64. Find the probability, p, that a patient will survive the flu. [0.975]
10. [2012/MJC/Prelim/II/12(a)]
The random variable X is the number of successes in n independent trials of an experiment in
which the probability of a success at a single trial is p. Denoting P(X = k) by pk , show that
pk+1
(n−k)p
pk = (k+1)(1−p) , k = 0, 1, 2, . . . , n − 1.
c 2017 Math Academy
www.MathAcademy.sg
2
[4]