Transcript Laws of Probability

I: Laws of Probbability
J: Independent Events
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Write what formulas we know on the board.
What other formulas are on the Mathematics
SL formula booklet (5.5 & 5.6)?
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The Addition Law
◦ For two events A and B, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵
Which means:
◦ 𝑃 𝒆𝒊𝒕𝒉𝒆𝒓 𝐴 𝒐𝒓 𝐵 𝒐𝒓 𝑏𝑜𝑡ℎ = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝒃𝒐𝒕𝒉 𝐴 𝒂𝒏𝒅 𝐵)
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Mutually Exclusive or Disjoint Events
◦ 𝐼𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝒎𝒖𝒕𝒖𝒂𝒍𝒍𝒚 𝒆𝒙𝒄𝒍𝒖𝒔𝒊𝒗𝒆 𝑒𝑣𝑒𝑛𝑡𝑠 𝑡ℎ𝑒𝑛 𝑃 𝐴 ∩ 𝐵 = 0
and so the addition law becomes:
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵
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Conditional Probability
◦ Given two events A and B, the conditional
probability of A given B is the probability that A
occurs given that B has already occurred.
◦ Written 𝐴 | 𝐵 and reads as “A given B”.
◦ If A and B are events then
P(𝐴 | 𝐵) =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
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Conditional Probability
◦ If A and B are events then
◦ It follows that:
P(𝐴 | 𝐵) =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
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In a class of 25 students, 14 like pizza and
16 like iced coffee. One student likes neither
and 6 like both. One student is randomly
selected from the class. What is the
probability that the student:
◦ Likes pizza?
◦ Likes pizza given that he or she likes iced coffee?
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In a class of 25 students, 14 like pizza and 16 like iced coffee. One student
likes neither and 6 like both. One student is randomly selected from the
class. What is the probability that the student:
◦ Likes pizza?
◦ Likes pizza given that he or she likes iced coffee?
The top shelf in a cupboard contains 3 cans of
pumpkin soup and 2 cans of chicken noodle soup.
The bottom shelf contains 4 cans of pumpkin soup
and 1 can of chicken noodle soup.
Jimmy is twice as likely to take a can from the
bottom shelf as he is from the top shelf. Suppose
Jimmy takes one can of soup without looking at
the label. Determine the probability that it:
a) is chicken.
b) was taken from the top shelf given that it
is chicken.
The top shelf in a cupboard contains 3 cans of pumpkin soup
and 2 cans of chicken noodle soup. The bottom shelf contains
4 cans of pumpkin soup and 1 can of chicken noodle soup.
Jimmy is twice as likely to take a can from the bottom shelf as
he is from the top shelf. Suppose Jimmy takes one can of soup
without looking at the label. Determine the probability that it:
a) is chicken.
b) was taken from the top shelf given that it is chicken.
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A and B are independent events if the
occurrence of each one of them does not
affect he probability that the other occurs.
𝑃 𝐴 𝐵) = 𝑃 𝐴 𝐵′) = 𝑃(𝐴).
Using 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝐵)𝑃(𝐵) we see that
A and B are independent events ↔
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃 𝐵 which we saw earlier.
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Definition of a mutually
exclusive event
If event A happens, then
event B cannot, or viceversa. The two events "it
rained on Tuesday" and "it
did not rain on Tuesday"
are mutually exclusive
events.
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When calculating
the probabilities for
exclusive events you add
the probabilities.
Mutually Exclusive Events
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Independent events
The outcome of event A, has
no effect on the outcome of
event B. Such as "It rained on
Tuesday" and "My chair broke
at work".
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When calculating the
probabilities for independent
events you multiply the
probabilities. You are
effectively saying what is the
chance of both events
happening bearing in mind
that the two were unrelated.
Independent Events
Clear as mud?