Introduction to Probability
Dr V. Babakov
Probability Outline
• Experimental Probability
• Theoretical Probability
• Equally Likely Outcomes
• Venn Diagram
• Probability tree
• Problems
Experimental Probability
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
• The conditions of the experiment have to remain stable
• As the number of trials keeps increasing, the experimental probability tends
towards the theoretical probability. To see this, the number trials should be
sufficiently large in number.
Example:
• If a dice is rolled 6000 times and the number '5' occurs 990 times, then the
experimental probability that '5' shows up on the dice is 990/6000 = 0.165
Theoretical Probability
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
For example,
the theoretical probability that the number '5' shows up on a dice when rolled is
1/6 = 0.167.
This is because of the 6 possible outcomes (dice showing '1', '2', '3', '4', '5', '6'), only
1 outcome (dice showing '5') is favourable.
Equally Likely Outcomes
When outcomes are equally likely, the Probability of a certain event is given by
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
Example:
What is the probability of getting a number less that 5 when a fair dice is tossing
once?
The event set is {1,2,3,4}. The sample set is {1,2,3,4,5,6}.
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
4
6
=
2
3
Theory of Probability
This table shows the distribution of daily sales of furniture at a particular store
Quantity
0
1
2
3
4
5
6
7
8
Probability
0.05
0.07
0.13
0.21
0.16
0.15
0.10
0.09
0.04
Calculate the probability of fewer than 5 sales
What is the probability of at least one sale?
Calculate the probability of 5 to 8 sales inclusive
Venn Diagram
A Venn diagram shows all possible logical relations between a finite collection of
sets.
The intersection (denoted as ∩) of two sets A and B is the set that contains all
elements of A that also belong to B (or equivalently, all elements of B that also
belong to A), but no other elements.
A
A∩B
B
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐴𝐴 ∩ 𝐵𝐵
Venn Diagram
The union (denoted by ∪) of a collection of sets is the set of
all distinct elements in the collection.
A
B
𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝐴𝐴 ∪ 𝐵𝐵
𝑃𝑃 𝐴𝐴 ∪ 𝐵𝐵 = 𝑃𝑃 𝐴𝐴 + 𝑃𝑃 𝐵𝐵 − 𝑃𝑃(𝐴𝐴 ∩ 𝐵𝐵)
We have to subtract that last term, because we added that
overlapping section twice when we took 𝑃𝑃 𝐴𝐴 + 𝑃𝑃(𝐵𝐵).
Venn Diagram
A class of 28 students were surveyed and asked if they ever had dogs or cats for
pets at home:
• 8 students said they had only ever had a dog.
• 6 students said they had only ever had a cat.
• 10 students said they had a dog and a cat.
• 4 students said they had never had a dog or a cat.
Probability tree
What is the probability of getting one boy and one girl?
P(BB) = 0.25
B
0.5
B
G
P(BG) = 0.25
B
P(GB) = 0.25
0.5
0.5
G
G
P(GG) = 0.25
Probability tree
On average, 1 in 20 people have a particular antibody mGF in their blood. The
probability tree shows the results when a machine tests the blood for mGF.
Calculate the probability that the machine gives incorrect result.
0.98
mGF
presents
Test positive
Test negative
0.05
0.03
Test positive
mGF
absent
Test negative
Problems
• Two coins are tossed, find the probability that two heads are obtained.
• A card is drawn at random from a deck of cards (52). Find the probability of
getting a queen.
• In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly.
What is the probability that it is neither red nor green?
• Suppose the probability that it rains in the next two days is 1/6 for tomorrow and
1/3 for the day after tomorrow. What is P (rain tomorrow, then rain day after
tomorrow)
• 0.25 1/13
1/3
1/18
One-to-one / small group advice for maths
Request a consultation with a Learning Advisor (Numeracy)
• Please email Dr Vitali Babakov
Visit the Learning Advisors website to:
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