Definition 1: Typically in the field of statistics we study data that results from experiments. An experiment can be considered to be a series of trials, each with a particular outcome. An event is a collection of outcomes corresponding to some result in the experiment. The number of outcomes in event E (i.e. the number of elements in set E) is written as |E|. The set of all possible outcomes is called the sample space, often designed S. An event is then simply a subset of the sample space. The probability P(E) of the event E is |E| / |S|, assuming S is not empty.
Example 1: Consider the simple experiment of tossing a coin twice. What is the probability that the coin comes up heads both times?
The sample space S = {HH, HT, TH, TT} and the required event E = {HH}. Thus the probability that the coin comes up heads both times is P(E) = |E| / |S| = ¼, or 25%.
Observation: We now state the fundamental properties of probability, using the usual set notation (see Sets for a quick review of this notation).
Property 1:
- 0 ≤ P(A) ≤ 1
- P(Ø) = 0
- P(S) = 1
- P(A′) = 1 – P(A), where A′ = S – A
- P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Proof: Simple consequences of Definition 1.
Example 2: Consider the experiment of drawing one card from a standard deck of 52 cards. What is the probability of drawing either a spade or face card?
There are 13 spades and 12 face cards, but 3 of these face cards are also spades, which we should not count twice. Thus, there are 13 spades and 9 non-spade face cards for a total of 22 cards out of 52. The probability is therefore 22/52. We now show how to calculate the result using Property 1e.
Let A = the event that a spade is drawn and B = the event that a face card (King, Queen, or Jack) is drawn. P(A) = 13/52, P(B) = 12/52 and P(A ∩ B) = 3/52. Thus the probability of drawing either a spade or face card is P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 13/52 + 12/52 – 3/52 = 22/52.
Definition 2: The probability that an event A occurs assuming that event B occurs is called the conditional probability of A given B and is denoted P(A|B).
Observation: By Definitions 1 and 2
Property 2:
- P(A|B) ∙ P(B) = P(A∩B) = P(B|A) ∙ P(A)
- P(A|B) = P(B|A) ∙ P(A) / P(B) called Bayes’ Theorem
- P(A) = P(A|B) ∙ P(B) + P(A|B′) ∙ P(B′) called the Law of Total Probability
Proof: The first assertion is a restatement of the last observation. The second assertion is a consequence of two applications of the first since
We now prove the third assertion. Since A = (A∩B) ∪ (A∩B′), by Properties 1b and 1e,
Now by Property 2a and 2b,
which proves the third assertion.
Example 3: Consider the experiment of picking two balls at random without replacement from a bag that contains 3 reds and 2 blacks. What is the probability that both balls are red?
Let A = a red ball is taken on the first draw and B = a red ball is taken on the second draw. The probability that the first draw is red is P(A) = 3/5. The probability that the second draw is red given the first draw is red is P(B|A) = 2/4 = ½. From Property 2a, we see that the probability that both draws are red is
Definition 3: Two events A and B are independent if P(A∩B) = P(A) ∙ P(B)
Property 3: Two events A and B are independent if and only if P(A) = P(A|B)
Proof: A and B are independent if and only if P(A∩B) = P(A) ∙ P(B), which by Property 2a is true if and only if P(A|B) ∙ P(B) = P(A) ∙ P(B), which in turn is true if and only if P(A|B) = P(A).
Observation: A and B are independent if B’s occurring (or not occurring) does not influence A’s occurring, i.e. it doesn’t increase or decrease the probability of A occurring. By Property 3, A and B are independent if and only if P(B|A) = P(B), and so it also follows that if A and B are independent then A’s occurrence does not influence B’s occurring either.
Example 4: Repeat the experiment from Example 3, but this time we put the ball picked on the first draw back in the bag before drawing a second ball (i.e. sampling with replacement).
Since P(B|A) = 3/5 = P(B), A and B are independent, it follows that
P(A∩B) = P(A) ∙ P(B) = 3/5 ∙ 3/5 = 36%.
Example 5: You have two bags, one containing 3 red and 2 black balls, the other containing 1 red, 1 blue, and 2 black balls. You pick a bag at random and then pick a ball from that bag at random. What is the probability that the ball picked is red?
Let A = event that the first bag is picked and let B = event that a red ball is drawn. By Property 2c,
P(B) = P(B|A) ∙ P(A) + P(B|A′) ∙ P(A′) = .6(.5) + .25(.5) = 42.5%.
Example 6: Suppose you roll a die 12 times. What is the probability that the number 1 will not appear on any of the throws? What is the probability that the number 1 will appear on at least one of the 12 throws?
The 12 throws represent 12 independent events. The probability of throwing a 1 on any single trial is 1/6 and so the probability of not throwing a 1 on any single trial is 1 – 1/6 = 5/6 (by Property 1d). Thus the probability of not throwing a 1 on any of the 12 throws is (5/6)12 = 11.2% (by Definition 3).
The probability that the number 1 will appear at least once is simply 1 – 11.2% = 88.8% (by Property 1d). This is equivalent to 1 – (1 – 1/6)12.
Help with statistical test assumptions on alley cropping systems
What sort of help do you need?
Charles
I am a college student and don’t know much about statistics. I need help to understand and complete my assignments, especially using Excel functions. Do you have any suggestions?
Thank you
Hello Beverly,
Clearly I intend the Real Statistics website as one way to learn statistics and to help with assignments.
Charles
You have a room with 50 people inside. What are the odds that at least two of them will have the same birthday?
Hello Leigh,
See https://en.wikipedia.org/wiki/Birthday_problem
Charles
As to example # 6 I take it that the probability of rolling a 1 all 12 times is (1/6) to the 12th. And that rolling a 1 exactly twice is 1/6*1/6*(5/6)to the 10th, correct?
But what if rolling the 1 instead of at least once in the twelve throws it was at least twice?
How is that computed?
Pete
Pete,
Actually rolling a 1 exactly twice has probability C(12,2) * (1/6)^2 * (5/6)^10. See the webpage on the Binomial Distribution.
Rolling a 1 at least twice has probability = 1 – P(0) – P(1) where P(0) = probability of rolling a 1 zero times = (5/6)*12 and P(1) = probability of rolling a 1 exactly one time = C(12,1) * (1/6) * (5/6)^11.
Charles