Introduction to Probability, Basic Overview - Sample Space, & Tree Diagrams

Main Ideas

• Probability measures how likely an event is to occur. It is written (P(A)), meaning “the probability of event (A).”

• For equally likely outcomes: [ P(A)=\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} ]

• A sample space is the set of all possible outcomes of an experiment.

• Probability ranges from 0 to 1:
  • (0): the event cannot happen.
  • (1): the event is certain to happen.
  • Multiply a probability by 100 to convert it to a percentage. For example, (0.3=30\%). Over many trials, this percentage describes an approximate expected frequency, not a guarantee for a particular set of trials.

Building Sample Spaces with Tree Diagrams

A tree diagram shows each possible result at each stage of an experiment.

• One coin flip: Sample space ({H,T}).

• Two coin flips: Each flip branches into (H) or (T), giving: [ {HH, HT, TH, TT} ]

• Three coin flips: Branching for a third flip gives eight outcomes: [ {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} ]

• For (n) coin flips, there are (2^n) possible outcomes because each flip has two possibilities.

Coin-Flip Probability Examples

Using the equally likely outcomes in each sample space:

• Two flips, at least one head: Favorable outcomes are (HH, HT, TH). [ P=\frac{3}{4}=0.75=75\% ]

• Three flips, at least two tails: Favorable outcomes are (HTT, THT, TTH, TTT). [ P=\frac{4}{8}=\frac12=50\% ]

• Three flips, exactly one tail: Favorable outcomes are (HHT, HTH, THH). [ P=\frac{3}{8}=0.375=37.5\% ]

Six-Sided Die Examples

The sample space for a fair six-sided die is ({1,2,3,4,5,6}).

• Roll a 2: One favorable outcome out of six: [ \frac16\approx 0.167\approx 16.7\% ]

• Roll a 3 or a 5: Two favorable outcomes out of six: [ \frac26=\frac13\approx 33.3\% ]

• Roll at most 4: Outcomes (1,2,3,4): [ \frac46=\frac23\approx 66.7\% ]

• Roll greater than 3: Outcomes (4,5,6): [ \frac36=\frac12=50\% ]

• Roll at most 5: Outcomes (1,2,3,4,5): [ \frac56\approx 83.3\% ]

General Method

1. List every possible outcome, using a tree diagram when the experiment has multiple stages.
2. Identify which outcomes satisfy the event in question.
3. Count the favorable outcomes and the total outcomes.
4. Divide the number of favorable outcomes by the total number of outcomes.
5. Simplify the fraction or convert it to a decimal or percentage if useful.

Speaker and Sources

• The Organic Chemistry Tutor is the sole instructor and narrator in the subtitles. The instructor also mentions the channel’s statistics playlist and other probability videos.