Problem 16E

A thermocouple placed in a certain medium produces readings within 0.1°C of the true temperature 70% of the time, readings more than 0.1°C above the true temperature 10% of the time, and readings more than 0.1°C below the true temperature 20% of the time.

a. In a series of 10 independent readings, what is the probability that 5 are within 0.1°C of the true temperature, 2 are more than 0.1 °C above, and 3 are more than 0.1 °C below?

b. What is the probability that more than 8 of the readings are within 0.1°C of the true temperature?

Solution :

Step 1 of 2:

Let denotes the number of readings.

Let is the readings within 0.10C of the true temperature 70% of the time.

So .

Let is the readings more than 0.10C above the true temperature 10% of the time.

So and

Let is the readings more than 0.10C below the true temperature 20% of the time.

So .

The total number of readings is 10.

Therefore MN(10,0.70,0.10,0.20)

Our goal is :

a). We need to find the probability that 5 are within 0.10C of the true temperature,

2 are more than 0.10C above and 3 are more than 0.10C below.

b). We need to find the probability that more than 8 of the readings are within 0.10C of the true

temperature.

a).

Now we have to find the probability that 5 are within 0.10C of the true temperature,

2 are more than 0.10C above and 3 are more than 0.10C below.

The multinomial probability formula is given by

P=

We consider ,and .

Then,

P(,,=

P(,,=(0.16807) (0.01) (0.003)

P(,,=0.00000134456

P(,,=25200.0000134456

P(,,=0.0338

Therefore P(,,is 0.0338.