To determine the probability that all 20 bulbs on the board will be switched on when the circuit is turned on, follow these steps:
1. Number of Bulbs and States:
Each of the 20 bulbs can be either on or off. This gives each bulb 2 possible states.
2. Total Possible Sequences:
For 20 bulbs, each having 2 possible states, the total number of possible sequences can be calculated as:
[tex]\[
2^{20}
\][/tex]
This equates to 1,048,576 different possible sequences of on/off states for the 20 bulbs.
3. Desired Sequence:
We are interested in one particular sequence where all the bulbs are on. There is only 1 such sequence out of the 1,048,576 possible sequences.
4. Calculating the Probability:
The probability of this desired sequence (all bulbs being on) is the ratio of the number of desired outcomes to the total number of possible outcomes, which is:
[tex]\[
\frac{1}{2^{20}} = \frac{1}{1,048,576}
\][/tex]
Thus, the probability that all the lights will be switched on is:
[tex]\[
\frac{1}{1,048,576}
\][/tex]
So the correct answer is:
C. [tex]\(\frac{1}{1,048,576}\)[/tex]
