Binomial distribution - IB Math HL and SL

1.      It has been claimed that in 60% of all solar-heat installations the utility bill is reduced by at least one-third.

 Accordingly, what are the probabilities that the utility bill will be reduced by at least one-third in

(a) four of five installations;

(b) at least four of five installations?

2.      If the probability is 0.05 that a certain wide-flange column will fail under a given axial load,

what are the probabilities that among 16 such columns

(a) at most two will fail;

(b) at least four will fail?

3.      Suppose that the algorithm, or robot reporter, typically writes proportion 0.65 of the stories on the site.

If 15 new stories are scheduled to appear on a website next weekend, find the probability that

(a) 11 will be written by the algorithm.

 (b) atleast 10 will be written by the algorithm

(c) between 8 and 11 inclusive will be written by the algorithm.

4.      Voltage fluctuation is given as the reason for 80% of all defaults in non-stabilized equipment in a plant.

Use the formula for the binomial distribution to find the probability that

 voltage fluctuation will be given as the reason for three of the next eight defaults.

5.      If the probability is 0.40 that steam will condense in a thin-walled aluminum tube at 10atm pressure,

Use the formula for the binomial distribution to find the probability that, under the stated conditions,

 steam will condense in 4 of 12 such tubes.

6.      A milk processing unit claims that, of the processed milk converted to powdered milk, 95% does not spoil.

Find the probabilities that among 15 samples of powdered milk

(a) all 15 will not spoil;

(b) at most 12 will not spoil;

(c) at least 9 will not spoil.

7.      The probability that the noise level of a wide-band amplifier will exceed 2dB is 0.05.Find the probabilities that

among 12 such amplifiers the noise level of

 (a) one will exceed 2 dB;

(b) at most two will exceed 2 dB;

(c) two or more will exceed 2 dB.

8.      If 6 of 18 new buildings in a city violate the building code, what is the probability that a building inspector

 who randomly selects 4 of the new buildings for inspection, will catch

 (a) none of the buildings that violate the building code?

 (b) 1 of the new buildings that violate the building code?

(c) 2 of the new buildings that violate the building code?

 (d) at least 3 of the new buildings that violate the building code?