A satellite relies on solar cells for its power and will...

A satellite relies on solar cells for its power and will operate provided that at least one of the cells is working. Cells fail independently of each other, and the probability that an individual cell fails within one year is 0.8. a) For a satellite with ten solar cells, find the probability that all ten cells fail within one year. b) For a satellite with ten solar cells, find the probability that the satellite is still operating at the end of one year. c) For a satellite with $n$ solar cells, write down the probability that the satellite is still operating at the end of one year. Hence, find the smallest number of solar cells required so that the probability of the satellite still operating at the end of one year is at least 0.95

A satellite relies on solar cells for its power and will operate provided that at least one of the cells is working. Cells fail independently of each other, and the probability that an individual cell fails within one year is 0.8.
a) For a satellite with ten solar cells, find the probability that all ten cells fail within one year.
b) For a satellite with ten solar cells, find the probability that the satellite is still operating at the end of one year.
c) For a satellite with $n$ solar cells, write down the probability that the satellite is still operating at the end of one year. Hence, find the smallest number of solar cells required so that the probability of the satellite still operating at the end of one year is at least 0.95

Key Concepts

Independent Events

This concept refers to random events whose outcomes do not affect each other. In the context of the satellite, each solar cell fails independently, meaning the failure of one cell does not influence the failure probability of another.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability that it does not occur. For example, the probability that at least one solar cell is working is one minus the probability that all cells fail.

Bernoulli Trials

Each solar cell's operation (working or failing) can be modeled as a Bernoulli trial where there are two outcomes: success (the cell works) with probability 0.2, and failure with probability 0.8. This framework is crucial for analyzing the overall satellite operation based on multiple independent cells.

Exponential Probability

When dealing with multiple independent Bernoulli trials, the probability of all events occurring (such as all cells failing) involves raising the individual probability to the power of the number of trials. This exponential relationship is important for calculating survival probabilities over multiple cells.

Inequalities and Logarithms in Probabilities

To determine the number of solar cells required to meet a specific operational probability, one often sets up an inequality using exponential probability expressions and then applies logarithms to solve for the unknown number of cells. This technique is key when adjusting system reliability thresholds.

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