**Unformatted text preview: **Probability
Rules of probability Probability
Probability is a measure of the likelihood
that a particular event will occur in any one
trial or experiment carried out under
prescribed conditions.
Each separate possible result is called an
outcome. Types of Probability (a) Empirical or experimental probability
is based on previous known results. The relative frequency of the number of
times the event has previously occurred
is taken as an indication of likely
occurrences in the future. Empirical or experimental
probability Example 1:
A random sample batch of 240
components is subjected to strict
inspection and 20 items are found to be
defective.
If any one component is picked at random,
chance of it being faulty is ‘20 in 240’ or ‘1
in 12’. Empirical or experimental
probability
If A is the event “a faulty component”, then
the probability of A happening, written as:
P(A) = 1 in 12 or 1/12 or 0.0833. Probability Expectation E is defined as the product of
the number of trials N and the probability
of the event A occurring in any one trial.
E = N.P(A) Expectation Therefore a run of 600 components from
the same machine would be expected to
contain 50 defectives:
1 = x 600 = x x = 50
12 600
12
This does not mean that there will definitely
be 50 defectives only that 50 are
expected.) Types of Probability
(b) Classical probability is based on a
consideration of the theoretical number of
ways in which it is possible for an event to
occur. Classical probability Example 2:
The probability that an event A might
occur is determined as follows:
P(A) = The number of ways in which
event A can occur divided by the total
number of all possible outcomes Classical probability Consider an unbiased die. When it is
rolled the total possible number of
outcomes is 6. Therefore the probability of throwing a 6 is
1/6 P(6) =1/6, while the probability of
throwing a 5 is also 1/6, P(5) =1/6. Classical probability If out of n possible outcomes of a trial, it is
possible for an event A to occur in x ways, and
for event A not to occur in y ways,
then n = x+ y.
And so P(A) = x = x
n
x+y A Classical probability If an event A is certain to happen every time
then x= n, y = 0, P(A) = n = 1.
n If an event A cannot possibly occur at any
time, then x = 0, y = n, P ( A) = 0 = 0. n Probability Mutually exclusive events are events
which cannot occur at the same time. Mutually non-exclusive events can
occur simultaneously. Probability Success or Failure:
Since we are concerned with the
probability of the occurrence of a particular
event, when it does occur it is recorded as
a success, and when it does not, a failure
is recorded. Probability
The probability of a success i.e. that of say
event A occurring is represented by P(A),
while the probability of a failure, that of event A
not occurring is represented by P ( A) Probability Also the sum of all possible probabilities is: P ( A) P ( A) 1 Independence Independent Events:
When the occurrence of one event does
not affect the probability of the occurrence
of the second event, the two events are
said to be independent. Independence Dependent events: When one event
affects the probability of the occurrence of
the second.
A pack of cards is shuffled and one card
is picked out, say an ace: P(A) = 4/52 = 1/13. If the card is not replaced after the first
draw the probability of picking an ace on
the second draw is now 3/51. Independence The probability of the second event is
dependent on the outcome of the first
event. This is denoted by P(B/A ), the probability
of B occurring given that A has happened. Independence Example 3: What is the probability of drawing two
aces from a pack of cards if the first ace is
not replaced ? P(Ace) = 4/52 =1/13 = 0.0769,
(No replacement). P(Second ace) = 3/51 = 0.0588
(correct to 4 decimal places). Probability Laws Multiplication Law:
Example 4:
Consider the probability of both events
A and B where event A represents
throwing a six when rolling a die, and
event B represents picking an ace from
a pack of cards. Probability Laws Roll both die and pick a card as one trial.
There are 6 possible outcomes for the die
and 52 possible outcomes from the cards
giving (6.52) outcomes altogether. There are 4 possibilities of obtaining a 6
and ace together, so the probability of
event A and event B occurring is: P(A and B) = 4 = 1 x 4 = 1 . 4
6 x 52 6 x 52 6 52 Probability Laws When A and B are independent events,
P(A and B) = P(A).P(B). Probability Laws Addition Law: Example 5:
Consider the probability of events A and B
where event A now represents the
probability of picking a 7 from a deck of
cards, and event B represents picking an
ace. Probability Laws If only one card is being picked then the
result can only be either an ace or a 7,
(these are mutually exclusive events).
The probability of event A or
event B occurring is: P(A or B) = 4 + 4 = 4 + 4 = 1 + 1 = 2
13 52 52 52 = P(A) + P(B). 13 13 Probability Laws When A and B are two mutually
exclusive events:
P(A or B) = P(A) +P(B). Probability Laws Example 6:
Consider the probability of picking a red
card or an ace. Let event A be the
probability of picking a red card, while
event B represents the probability of
picking an ace.
Both events could occur simultaneously
(not mutually exclusive events). However
we are only interested in the probability of
either event A or event B not both. Probability Laws P(A or B) = 26 + 4 = 26 + 4
52 52 52 This follows the same trend as before but we
must take into account the probability of a
red ace which we do not want! Therefore P(A or B) = 26 + 4 - 2 = 28 = 7
52 52 52 52 = P(A) or P(B) - P(A and B). 13 Probability Laws When A and B are not mutually
exclusive events,
P(AorB) = P(A) + P(B) -P(A).P(B) Probability Laws Example7:
Consider the probability of scoring a multiple of 3
i.e. (3 or 6) when a die is rolled.
Let this be event A. P(A) = 1/6 +1/6 = 2/6.
Now consider the probability of throwing a
multiple of 2 i.e. (2, 4 or 6).
Let this be event B.
P(B) = 1/6 + 1/6 +1/6 = 3/6. Probability Laws Now what is the probability of scoring a
multiple of 2 or 3? Again these events are not mutually
exclusive since 6 is a multiple of both 2
and 3. Therefore the probability of scoring
one of these sixes must be removed: Probability Laws P(A or B) = P(A) or P(B) - P(A and B)
= 2/6 + 3/6 - (2/6.3/6)
= 2/6 + 3/6 - 6/36
= 2/6 + 3/6 - 1/6
= 2/3 Probability Laws Summary: Multiplication Law: ‘AND’ means multiply. Addition Law: ‘EITHER /OR’ means add Mutually Exclusive: Cannot happen at
same time ...

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- Spring '14
- Probability, Playing card