Home » Basic probability (Page 8)
Category Archives: Basic probability
Dice Rolls – Max and Min
Suppose two 4-sided dice are rolled. Find the probabilities of the following events:
- the maximum of the two numbers rolled is less than or equal to 2;
- the maximum of the two numbers rolled is less than or equal to 3;
- the maximum of the two numbers rolled is equal to 3;
- Repeat part c for the maximum equal to 1, 2, and 4.
- If M is the maximum of the two numbers, then
\[\mathbf{P}(M = 1) + \mathbf{P}(M = 2) + \mathbf{P}(M = 3) + \mathbf{P}(M = 4) = 1 \]
check that your answers for 3) and 5) satisfy this relationship.
Random Letters
Suppose a word is picked at random from this sentence. Find:
- the outcome space for this random experiment
- the chance the word has at least 4 letters;
- the chance that the word contains at least 2 vowels (a,e,i,o,u)
- the chance that the word contains at least 4 letters and at least 2 vowels.
- What is the distribution of the length of the word picked ?
- What is the distribution of the number of vowels in the word ?
(Based on [Pitman p.9 #2 and p. 31 #6)
Unions and intersections of events
Let \(A\) and \(B\) be any two events. Define the new events \(C\), \(\hat A\), and \(\hat B\) by \(C=A\cap B\), \(\hat A=A \cap B^c\), and \(\hat B = B \cap A^c\) where \(A^c\) is the compliment of \(A\) and \(B^c\) is the compliment of \(B\).
- Argue that \(A \cup B = \hat A \cup \hat B \cup C\) and that all three sets are mutually disjoint. i.e. \(\hat A\cap C = \emptyset\), \(\hat B\cap C = \emptyset\), and \(\hat A\cap \hat B = \emptyset\).
- Show that \(\mathbf{P}(A)= \mathbf{P}(\hat A) + \mathbf{P}(C)\) and \(\mathbf{P}(B)= \mathbf{P}(\hat B) + \mathbf{P}(C)\) .
- Show that \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B) – \mathbf{P}(A \cap B)\).
Sums of normals
- Consider a normal random variable \(X\) with mean \(\mu_1\) and standard deviation \(\sigma_1\)
- Consider a normal random variable \(Y\) with mean \(\mu_2\) and standard deviation \(\sigma_2\).
Assume that \(X\) and \(Y\) are independent and define \(Z=X+Y\)
- What is the distribution of \(Z\) ?
- What is the mean and variance of \(Z\) ?
- (**) If we now assume that they are not independent, but still normal as described above, what can you say ?
Standardized Random Variables
Consider a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). Define a new random variable \(Y\) by
\[Y=\frac{X-\mu}{\sigma}\,.\]
- Show that \(Y\) has mean 0 and variance 1.
- Show that if \(a \) is some number \[\mathbf{P}( Y > a) = \mathbf{P}( X > \mu + a\sigma )\]
