Chapter 3: Q3-56E (page 193)

Two fair coins are tossed, and the following events are defined:
A: [Observe one head and one tail.]
B: [Observe at least one head.]
a. Define the possible sample points and assign probabilities to each.
b. Draw a Venn diagram for the experiment, showing the sample points and events A and B.
c. Find P(A), P(B) and $P (A \cap B)$ .
d. Use the formula for conditional probability to find P (A/B)and P (B/A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.

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Answer

Step by step solution

Probability is the study of prior records or the quantity and kind of probable outcomes to predict an event's result.

$Sample points =[(H, H), (H, T), (T, H), (T, T)] P (A)= [(H, T), (T, H)] P (B)= [(H, H), (H, T), (T, H)]$

Here are all the sample points and events of A and B.

$\begin{matrix} P (A) = \frac{2}{4} \\ = \frac{1}{2} \\ = 0.5 \end{matrix}$

$\begin{matrix} P (B) = \frac{3}{4} \\ = 0.75 \end{matrix}$

$\begin{matrix} P (A \cap B) = \frac{2}{4} \\ = \frac{1}{2} \\ = 0.5 \end{matrix}$

Hence, the required probabilities are 0.5, 0.75 & 0.5.

$\begin{matrix} P (A/B) = \frac{P (A \cap B)}{P (B)} \\ = \frac{0.50}{0.75} \\ = 0.67 \end{matrix}$

$\begin{matrix} P (B/A) = \frac{P (A \cap B)}{P (A)} \\ = \frac{0.50}{0.50} \\ = 1 \end{matrix}$

Hence, the required probabilities are 0.67, 1.

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Network Type/Number of Channels	Number of Jamming Attacks
WLAN / Single	31
WSN / Single	13
AHN / Single	8
WLAN / Multi	14
WSN / Multi	9
AHN / Multi	5
TOTAL	80