An event has 4 possible outcomes with probabilities 1/2, 1/4, 1/8, 1/16. What will be the rate of information if there are approximately 24 outcomes/second possible?
39 bits/sec
To determine the rate of information, we first need to calculate the average information per outcome, which is also known as entropy. The information content of an individual outcome is related to its probability. The rate of information is then found by multiplying this average information by the number of outcomes occurring per second.
Information Content Calculation
The information content $I(P_i)$ for an outcome with probability $P_i$ is given by the formula:
$$I(P_i) = -\log_2(P_i)$$
Let's calculate the information content for each of the four given outcomes:
- For the first outcome with probability $P_1 = \frac{1}{2}$: $$I_1 = -\log_2\left(\frac{1}{2}\right) = -\log_2(2^{-1}) = -(-1)\log_2(2) = 1 \text{ bit}$$
- For the second outcome with probability $P_2 = \frac{1}{4}$: $$I_2 = -\log_2\left(\frac{1}{4}\right) = -\log_2(2^{-2}) = -(-2)\log_2(2) = 2 \text{ bits}$$
- For the third outcome with probability $P_3 = \frac{1}{8}$: $$I_3 = -\log_2\left(\frac{1}{8}\right) = -\log_2(2^{-3}) = -(-3)\log_2(2) = 3 \text{ bits}$$
- For the fourth outcome with probability $P_4 = \frac{1}{16}$: $$I_4 = -\log_2\left(\frac{1}{16}\right) = -\log_2(2^{-4}) = -(-4)\log_2(2) = 4 \text{ bits}$$
We can summarize these results in a table:
| Outcome | Probability ($P_i$) | Information Content ($I_i$) |
|---|---|---|
| 1 | $\frac{1}{2}$ | 1 bit |
| 2 | $\frac{1}{4}$ | 2 bits |
| 3 | $\frac{1}{8}$ | 3 bits |
| 4 | $\frac{1}{16}$ | 4 bits |
Average Information (Entropy) per Outcome
The average information per outcome, or entropy ($H$), is calculated by summing the product of each outcome's probability and its information content. The formula for entropy is:
$$H = \sum_{i=1}^{n} P_i \cdot I_i$$
Using the values we calculated:
$$H = \left(\frac{1}{2} \times 1\right) + \left(\frac{1}{4} \times 2\right) + \left(\frac{1}{8} \times 3\right) + \left(\frac{1}{16} \times 4\right)$$
$$H = \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16}$$
Simplify the fractions:
$$H = \frac{1}{2} + \frac{1}{2} + \frac{3}{8} + \frac{1}{4}$$
To add these fractions, find a common denominator, which is 16:
$$H = \frac{8}{16} + \frac{8}{16} + \frac{6}{16} + \frac{4}{16}$$
$$H = \frac{8 + 8 + 6 + 4}{16} = \frac{26}{16} = \frac{13}{8} \text{ bits/outcome}$$
Rate of Information Calculation
The rate of information is found by multiplying the average information per outcome (entropy) by the number of outcomes possible per second. The question states there are approximately 24 outcomes/second possible.
$$\text{Rate of Information} = H \times (\text{Outcomes per second})$$
$$\text{Rate of Information} = \frac{13}{8} \text{ bits/outcome} \times 24 \text{ outcomes/second}$$
$$\text{Rate of Information} = 13 \times \left(\frac{24}{8}\right) \text{ bits/second}$$
$$\text{Rate of Information} = 13 \times 3 \text{ bits/second}$$
$$\text{Rate of Information} = 39 \text{ bits/second}$$
Final Information Rate Result
The calculated rate of information for the given event and outcome rate is 39 bits/sec.
Important Questions from Types of Probability
For the joint density f xy (x, y) = x 2 + Cy; 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, the value of constant C is:
Let A, B, C be 3 independent events such that P(A) = \(\frac{1}{3}\) , P(B) = \(\frac{1}{2}\) , P(C) = \(\frac{1}{4}\) , then probability of exactly 2 events occurring out of 3 events is:
If f(x) is a probability density on the real line, then which of the following is NOT a valid probability density?
A die is tossed three times, What is the probability of getting an odd number at least once ?
The probability of student A passing an exam is 2/7 and that of B passing is 5/7. If these probabilities are independent, what is the probability that only B passes the examination
