Instructions

• This section covers the concepts listed below. 
• For each concept, there is a conceptual video explaining it followed by videos working through examples. 
• When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page. 

Concepts

• Kolmogorov’s Axioms
• Discrete Uniform Probability Spaces
• Complement Rule
• Monotonic Property and Inclusion & Exclusion

Links & Resources

Examples

1. Suppose you have a biased coin in which heads is 3 times more likely to occur than tails. If you flip this coin, what is the probability of getting heads? tails?
   Reveal Answer

   \(P(\{H\})=\dfrac{3}{4} \mbox{ and } P(\{T\})=\dfrac{1}{4}\)

   Watch Video Solution

   ---

2. Flip a fair coin until a tail comes up. Establish the probabilistic model for such a random
   experiment.
   Reveal Answer

   See video for further explanation.

   Watch Video Solution

Self-Assessment Questions

1. What is a probability space? Can you state the axioms?

   ---

2. What is the difference between an outcome and an event in a probabilistic model? To which one do we assign probability?

   ---

3. Does $P(A)=0$ imply $A=\mathrm{\varnothing }$? Can you provide a counterexample, if not?

   ---

4. Can you describe ${\displaystyle \bigcup _{n=1}^{\mathrm{\infty }}[\frac{1}{n},1]}$ and ${\displaystyle \bigcap _{n=1}^{\mathrm{\infty }}(-\frac{1}{n},\frac{2}{n})}$?

   ---

5. Does ${\displaystyle \cdot \bigcup _{n=1}^{\mathrm{\infty }}[\frac{1}{n},1]}$ make sense?

Examples

1. We roll a pair of fair dice. What is the probability of getting a sum of 10?
   Reveal Answer

   \(\dfrac{1}{12}\)

   Watch Video Solution

Self-Assessment Questions

1. Can we have an infinite discrete uniform probability space?

   ---

2. Does "rolling a pair of fair dice and recording the product of the two numbers as the outcome" define a uniform probability space?

Examples

1.  Roll a fair die four times. What is the probability that some number appears more than once?
   Reveal Answer

   \(\dfrac{13}{18}\)

   Watch Video Solution

Self-Assessment Questions

1. What does the Complement Rule say, and why is it useful?

   ---

2. What is the complement of the following event in the random experience of rolling two dice: $A=$ the event of getting a sum of at least 4? Determine all the outcomes of $A^c$.

Examples

1. 15% of the population in a town is blond, 25% has blue eyes, and 2% is blond with blue eyes. What is the probability that a randomly chosen individual is not blond and does not have blue eyes?
   Reveal Answer

   62%

   Watch Video Solution

   ---

2. There are 30 red balls, 20 green, and 10 yellow balls in an urn. Draw 7 balls without replacement. What is the probability that exactly 2 red or exactly 3 yellow balls are in the sample?
   Reveal Answer

   \(P(A\cup B) = \displaystyle \frac{ {30 \choose 2 } {30 \choose 5}}{{60 \choose 7}}+ \frac{\binom{10}{3} \binom{50}{4}}{\binom{60}{7}} - \frac{\binom{30}{2} \binom{10}{3} \binom{20}{2}}{\binom{60}{7}}\)

   Watch Video Solution

Self-Assessment Questions

1. State the monotonic property of probability.

   ---

2. When do we have $P(A\cup B)=P(A)+P(B)$?