STA215H5 Lecture Notes - Lecture 12: Fair Coin, Sample Space
STA215; Chapter 12
What is probability?
● Flip a coin. Can you predict the outcome? It’s hard to guess the outcome based on just
one flip because the outcome is random. If it is a fair coin though, you can predict the
proportion of heads you’re likely to see in the long-term. To do this, we need to talk
about the probability of different outcomes
● Probability is a numerical measure of the likelihood that an event will occur. It is
expressed as a number between 1 and 0.
○ An event with probability of 1 can be considered a certainty. For example, the
probability of a coin toss resulting in either ”heads” or ”tails” is 1, because there
are no other options, assuming the coin lands flat.
○ An event with probability of 0.5 can be considered to have equal odds of
occurring or not occurring. For example, the probability of a coin toss resulting in
”heads” is 0.5, because the toss is equally as likely to result in ”tails.”
○ An event with a probability of 0 can be considered an impossibility. For example,
the probability that the coin will land (flat) without either side facing up is 0,
because either ”heads” or ”tails” must be facing up.
● Example; Rolling a fair die.
○ The list of all possible outcomes are = 1, 2, 3, 4, 5, 6 .
The probability of rolling a 4 = ⅙
● Definitions;
○ The sample space S of a random phenomenon is the set of all possible
outcomes.
○ Each occasion when we observe a random phenomenon is called a trial.
○ At each trial, we note the value of the random phenomenon and called that the
trial’s outcome.
○Outcome: the value of the random phenomenon at each trial
○ An event is an outcome or a set of outcomes of a random phenomenon. That is,
an event is a subset of the sample space.
● Example;
○ In each of the following situations, describe the sample space S for the random
phenomenon.
■ a) A basketball player shoots three free throws. You record the sequence
of hits and misses.
● S = {(HHH), (HHM), (MHH), (HMH), (MHM), (MMH), (HMM),
(MMM)}
■ b) A basketball player shoots three free throws. You record the number of
baskets she makes.
● The player could either get 0, 1, 2, or 3 baskets
● S = {0, 1, 2, 3}
Document Summary
It"s hard to guess the outcome based on just one flip because the outcome is random. If it is a fair coin though, you can predict the proportion of heads you"re likely to see in the long-term. To do this, we need to talk about the probability of different outcomes. Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 1 and 0. An event with probability of 1 can be considered a certainty. For example, the probability of a coin toss resulting in either heads or tails is 1, because there are no other options, assuming the coin lands flat. An event with probability of 0. 5 can be considered to have equal odds of occurring or not occurring. For example, the probability of a coin toss resulting in. Heads is 0. 5, because the toss is equally as likely to result in tails.
