The learning outcomes of this topic are:
- recall the rules of simple probability
- use key probability distributions (Binomial distribution, Poisson distribution, Exponential distribution, Normal distribution)
- calculate z-scores
This topic will cover:
- Simple probability revision
- Probability distributions
- Standard scores (z-scores)
2. This topic will cover:
◦ Simple probability revision
◦ Probability distributions
◦ Standard scores (z-scores)
3. By the end of this topic students will be able
to:
◦ recall the rules of simple probability
◦ use key probability distributions:
Binomial distribution
Poisson distribution
Exponential distribution
Normal distribution
◦ calculate z-scores
5. ◦ Sample space
Set of all possible outcomes
◦ Event
One or more outcomes
6. E2
◦ Mutually exclusive events
events that cannot occur
together
P(E1 or E2) = P(E1) + P(E2)
◦ Non-mutually exclusive events
P(E1 or E2) = P(E1) + P(E2) -
P(E1 ∩ E2)
E1
E1 E2E1∩ E2
7.
8. ◦ Discrete
Number of customers per hour
Therefore seek model Probability Mass
Functions that give P(X = x)
Number Frequency
Empirical
Probability
0 10 0.0833
1 17 0.1417
2 42 0.3500
3 34 0.2833
4 12 0.1000
5 5 0.0417
120 1
9. ◦ Continuous
height of customers
therefore seek model probability density
functions that lead to P(xl < X < xh)
Height Frequency
Empirical
Probability
163 -165 1 0.005
166 -168 4 0.020
169 -171 14 0.070
172 -174 29 0.145
175 -177 44 0.220
178 -180 46 0.230
181 -183 35 0.175
184 -186 18 0.090
187 -189 7 0.035
190 -192 2 0.010
200 1
10. ◦ Sample space
Set of all possible outcomes
◦ Event
One or more outcomes
◦ Mean (of a random variable)
𝜇 =
𝑓𝑖 𝑥𝑖
𝑁
⟶ 𝜇 = 𝑝𝑖 𝑥𝑖
◦ Standard Deviation (of a random variable)
𝜎 =
𝑓𝑖 𝑥𝑖 − 𝜇 2
𝑁
⟶ 𝜎 = 𝑝𝑖 𝑥𝑖 − 𝜇 2
11. ◦ A TRIAL has two possible outcomes
P(success) = p, P(failure) = 1 - p
Pass or fail training, medical treatment works or
not, aeroplane engine works or not, meet SLA or not
etc.
◦ Number of such trials, n, takes place
10 workers undergo training how many might pass?
1000 patients are treated, how many may recover?
4 working engines on aeroplane, how many will fail?
◦ Q ~ B(n, p)
17. ◦ Rare event A in background of not A
Large n and small p, np = l
◦ Probability of a number of independent, randomly
occurring successes (or failures) within a specified
interval
Number of customers arriving at end of queue
Number of print errors per area
Number of machine breakdowns per year
◦ A ~ Po (l)
25. ◦ Either tables or software
can then give partial
areas under the curve
which indicate
probabilities of
particular values of z
occurring.
P(Z < z)
P(Z > z)P(0 < Z < z)
27. By the end of this topic students will be able
to:
◦ recall the rules of simple probability
◦ use key probability distributions;
Binomial distribution
Poisson distribution
Exponential distribution
Normal distribution
◦ calculate z-scores