○P ROBABILITY○
calculating probability
Random event – event where the result is random, not known before the event
Eg. tossing a coin, rolling the dice
Sample space (S or 𝛀) – representation of the complete set of all possible outcomes from
an experiment. It can be a list, a table or a diagram.
Sample space consist of elementary events, for example:
tossing a coin: S = {H, T}
rolling the dice: Ω = {1,2,3,4,5,6}
Types of events (on the example of rolling the dice):
● elementary event - exact possible outcome, e.g. rolling the 6
● event - thing that can happen generally, e.g. event A = rolling an odd number
● certain event - event that will surely happen, e.g event B = {1,2,3,4,5,6}
● impossible event (Ø) - event that surely won’t happen, e.g. rolling negative number
● complementary event - the event with opposite outcome to the other event,
e.g. event C’ = {1,2,3,4} if event C = {5,6}
Probability – chance of the event occurring
If the sample space consist of equally likely elementary events then (“classic approach”):
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑛(𝐴)
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
so 𝑃(𝐴) 𝑛(𝑆)
Properties of probability of an event [note: 1 = 100%]
● 0 ≤ P(A) ≤ 1
● P(A) + P(A’) = 1
Methods of representing the sample space visually:
example - rolling two dices:
calculating probability
Random event – event where the result is random, not known before the event
Eg. tossing a coin, rolling the dice
Sample space (S or 𝛀) – representation of the complete set of all possible outcomes from
an experiment. It can be a list, a table or a diagram.
Sample space consist of elementary events, for example:
tossing a coin: S = {H, T}
rolling the dice: Ω = {1,2,3,4,5,6}
Types of events (on the example of rolling the dice):
● elementary event - exact possible outcome, e.g. rolling the 6
● event - thing that can happen generally, e.g. event A = rolling an odd number
● certain event - event that will surely happen, e.g event B = {1,2,3,4,5,6}
● impossible event (Ø) - event that surely won’t happen, e.g. rolling negative number
● complementary event - the event with opposite outcome to the other event,
e.g. event C’ = {1,2,3,4} if event C = {5,6}
Probability – chance of the event occurring
If the sample space consist of equally likely elementary events then (“classic approach”):
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑛(𝐴)
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
so 𝑃(𝐴) 𝑛(𝑆)
Properties of probability of an event [note: 1 = 100%]
● 0 ≤ P(A) ≤ 1
● P(A) + P(A’) = 1
Methods of representing the sample space visually:
example - rolling two dices:
