PROBABILITY AND STATISTICS

PROBLEM 2.

The electronic telemetry system aboard a spacecraft transmits the data of spacecraft motion in the x, y, and z directions. The system consists of three motion sensors, a signal conditioner, and a transmitter. The probability of failure for each motion sensor and for the signal conditioner is 0.0001. The probability of failure for the transmitter is 0.001. Assuming that component failures are independent events and that the failure of any component will render the telemetry system inoperative, compute the probability of a spacecraft telemetry success.

Solution: The probability of success is equal to 1 minus the probability of failure. Therefore, the probability of success for each sensor and the signal conditioner is

P = I - 0.0001 = 0.9999.

Similarly, the probability of success for the transmitter is

P = 1 - 0.001 = 0.999.

The probability of success for the telemetry system is the product of probabilities of success for each component; that is,

P = (0.9999)^4(0.999) = 0.9986.

The signals transmitted by a spacecraft telemetry system are in the form of pulses imposed on a radio beam, which can be interpreted as binary digits. For example, a signal fragment might be represented as 010110 in which case the presence of a pulse is read as 1 and the absence of a pulse as 0. Each possible representation of a 0 or a 1 is called a "bit."

For a variety of reasons, equipment errors can cause a O to be transmitted instead of a 1, or vice versa. As a result, error-detecting codes have been developed to improve data reliability. All such codes are based on transmitting extra bits that can be used to determine whether errors are present and even, for the more sophisticated codes, where the errors are. Transmitting these extra bits, however, means that fewer message-carrying bits can be sent in a given unit of time, and so transmission reliability must be traded against transmission efficiency. Probability theory plays an important part in weighing the trade-offs.

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