In deterministic logic, the statement “$A$ implies $B$” is equivalent to its contrapositive, “not $B$ implies not $A$”. In this problem we will consider analogous statements in probability, the logic of uncertainty. Let $A$ and $B$ be events with probabilities not equal to 0 or 1. (a) Show that if $P(B|A) = 1$, then $P(A^c|B^c) = 1$. Hint: Apply Bayes’ rule and LOTP. (b) Show however that the result in (a) does not hold in general if $=$ is replaced by $\approx$. In particular, find an example where $P(B|A)$ is very close to 1 but $P(A^c|B^c)$ is very close to 0. Hint: What happens if A and B are independent?

Using set notation, write out the sample space for each of the following random experiments. (a) A coin is tossed four times in a row. The observation is how the coin lands (H or T) on each toss. (b) A student randomly guesses the answers to a four-question true-or-false quiz.The observation is the student's answer (T or F) for each question. (c) A coin is tossed four times in a row. The observation is the percentage of tosses that are heads. (d) A student randomly guesses the answer to a four-question true-or-false quiz. The observation is the percentage of correct answers in the test.

The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called _______ of _______ .

Find its zeroes, i any.

$$g ( x ) = \frac { x ^ { 3 } + 2 x } { x ^ { 2 } - 4 }$$

The screens used for a certain type of cell phone are manufactured by 3 companies, A, B, and C. The proportions of screens supplied by A, B, and C are 0.5, 0.3, and 0.2, respectively, and their screens are defective with probabilities 0.01, 0.02, and 0.03, respectively. Given that the screen on such a phone is defective, what is the probability that Company A manufactured it?

Find the limit by rewriting the fractions first.

$$\lim _ { ( x , y ) \rightarrow ( 4,3 ) } \frac { \sqrt { x } - \sqrt { y + 1 } } { x - y - 1 }$$

Where

$${x \neq y + 1}$$

Why was a distinctly new form of federalism created by the Founders?

Three electromagnetic waves travel through a certain point P along an x axis. They are polarized parallel to a y axis, with the following variations in their amplitudes. Find their resultant at P.

$$E_1 = (10.0 μV/m) sin[(2.0 × 10^14 rad/s)t]$$

$$E_2 = (5.00 μV/m) sin[(2.0 × 10^14 rad/s)t + 45.0°]$$

$$E_3 = (5.00 μV/m) sin[(2.0 × 10^14 rad/s)t - 45.0°]$$

What is a laplace transform?

You know that

$$\overline { A B } \cong \overline { X Y } , \overline { B C } \cong \overline { Y Z } , \text { and } \overline { C A } \cong \overline { Z X }.$$

By what theorem or postulate can you conclude that

$$\triangle A B C \cong \triangle X Y Z?$$

For the quadratic equation

$$x^2+a=0$$

, determine whether each value of a will result in two rational solutions. Explain.

$$\frac { 1 } { 2 }$$

Use the information given about the angles a and b to find the exact value of: $\sin (\alpha-\beta)$. $\sin \alpha=-\frac{2}{3}, \pi<\alpha<\frac{3 \pi}{2} ; \cos \beta=-\frac{2}{3}, \pi<\beta<\frac{3 \pi}{2}$

In a certain Precalculus class, there are 18 freshmen and 15 sophomores. Of the 18 freshmen, 10 are male, and of the 15 sophomores, 8 are male. Find the probability that a randomly selected student is: A freshman or female.

State whether each number is divisible by 2, 3, 5, 9, 10, or none.

$$30$$