1MA01: Mathematical Methods Tutorial Sheet
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1. Write down the addition and multiplication rules for probabilities.
addition P (A and B) = P (A) + P (B)
multiplication P (AorB) = P (A)P (B|A).
Consider four job openings for which there are eight applicants. Five
are men and three are women. If all candidates are equally likely to be
chosen what is the probability that no women are chosen?
P (no women chosen) = P (all men chosen)
5 4 3 2
× × ×
=
8 7 6 5
1
=
14
= 0.07142
or approximately 7% probability that no women are chosen.
2. Consider a ten-sided die, each side labelled with a number from one to
ten. The die is rolled once. Find
(a) P (4 or 8): the probability of rolling a 4 or an 8
1
1
P (4) = 10
and P (8) = 10
so P (4 or 8) = P (4) + P (8) =
2
10
=
1
5
(b) P (9 or less than 3): the probability of rolling a 9 or a number less
than 3.
1
2
P (9) = 10
and P (less than 3) = 10
(since there are 2 numbers less
than 3: 1 and 2). Now
3
P (9 or less than 3) = P (9) + P (lessthan3) = 10
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Sinéad Ryan, ryan@maths.tcd.ie, see also http://www.maths.tcd.ie/˜ryan/123.html
(c) P (5 or an odd number): the probability of rolling 5 or an odd
number
5
1
and P (odd number) = 10
since there are 5 odd numbers
P (5) = 10
(1,3,5,7,9) and 10 possible outcomes. Now
1
1
5
1
P (5 or an odd number) = P (5) + P (odd) − 10
= 10
+ 10
− 10
= 21 .
Note that you have to subtract off 1/10 since the probability of
rolling 5 is included in both.
(d) P (odd or prime number): the probability of rolling an odd number or a prime number
Here the odd numbers are: 1,3,5,7,9; and the prime numbers are:
2,3,5,7. Notice that 3,5,7 are both odd and prime and so will
appear in both probabilities. So,
4
5
and P (prime) = 10
giving,
P (odd) = 10
4
3
5
+ 10
− 10
= 35 .
P (odd or prime number) = 10
3. Write down the formula for computing success in a binomial experiment.
P (X = k) =
n
k
!
pk q n−k .
A survey finds that 30% of teenage consumers receive their spending
money from part-time jobs. If five teenagers are selected at random
from a large group, find the probability that at least three of them will
have part-time jobs.
The probability that at least three have jobs is the sum of the probability
that 3 have a job (P (X = 3)) and the probability that 4 have a job
(P (X = 4)) and the probability that 5 have a job (P (X = 5)).
So we need P (X = 3) + P (X = 4) + P (X = 5)
Now, with p = 0.3 since 30% have a job so q = 0.7 and n = 5. Then
plugging into the formula above: P (X = 3) = 0.1323, P (X = 4) =
0.02835, P (X = 5) = 0.00243, so probability = 0.16306, ie about 16%.
4. Consider the following experiments: State whether or not they are
binomial experiments and justify your answers.
(a) Anne rolled a 6-sided number cube until a 3 appeared.
(b) You are at a fair, playing “pop the balloon” with 6 darts. There
are 20 balloons. 10 of the balloons have a ticket inside that say
“win” and 10 have a ticket that says “lose”.
(c) Asking 200 people if they watch RTE news.
(d) You take a survey of 50 traffic lights in Dublin, at 3 p.m., recording
whether the light was red, green, or yellow at that time.
(e) Flip a coin 30 times to see how many heads you get.
(a) Not binomial. Not a fixed number of trials
(b) Not binomial. The probability of success (getting a “win”) changes
each time.
(c) Binomial. Fixed number of trials (20), probability of success constant; Outcome is success or failure (watches or doesn’t watch), experiment counts the number of successful outcomes (ie how many
people watch RTE)
(d) Not binomial. 3 possible outcomes (red, green or yellow) not “success” or “failure”.
(e) Binomial. Fixed number of trials (30), probability of success
(heads) constant ie 1/2, outcome is success or failure each time
and experiment counts the number of successes.