1
You have a fair coin that you toss eight times. What is the probability that you’ll get no more than seven heads?
- Probability is 1-chance of getting 8 heads
- \(\frac{1}{2}^8 = \frac{1}{256}\)
2
You have a fair coin that you toss eight times. What is the probability that you’ll get exactly seven heads?
- 7 different ways to get exactly seven heads
- \(\frac{1}{2}^8 = \frac{1}{256}\) chance for each way
- \(7 \cdot \frac{1}{256} = \frac{7}{256}\)
3
Let P(X) = 0.2, P(Y) = 0.4, P(X|Y) = 0.5. What is P(Y|X)?
- Bayes rule is \(P(A|B) = \frac{P(B|A)\cdot P(A)}{B}\)
- \(P(Y|X) = \frac{0.5 \cdot 0.4}{0.2} = 1\)
4
Let P(X) = 0.2, P(Y) = 0.4. If P(X|Y) = 0.2, what can you say about X & Y?
- Bayes rule can be used again, but this can also be reasoned out
- \(P(X) = 0.2 = P(X|Y)\)
- The probability of X is the same as the probability of X given Y
- Y has no relationship to X
- They are independent
5
Which of these numbers cannot be a probability?
0, 1.0, 1.5, 0.5
Probabilities are given as a chance of an event occurring against some other condition. This means that probabilities cannot be greater than 1 or less than 0.
- 1.5
6
A die is rolled and a coin is tossed simultaneously. What is the probability of getting an even number on the die and a head on the coin?
- Probability of even numbers on a 6 sided die is 1/2
- probability of a head on a coin flip is 1/2
- \(\frac{1}{2}^2 = \frac{1}{4}\)
7
Let \(f(x) = x^2\) What is its integral and differential?
- Power rule of integration \(\int x^2dx = \frac{x^3}{3} + c\) Where c is an unknown constant.
- Power rule of derivation \(\frac{df}{dx}x^2 = 2x\).
8
Let A be a 3x4 matrix of the following format
\(A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 1 \end{bmatrix} \)
What is the rank of A?
- Compute echelon form
- \(A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 1 \end{bmatrix} \rightarrow \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
- 2 non zero rows, rank=2