# If a random sample of 12 of the new type of tire are tested, what is the probability that at least 2 of them will last longer than 75000km?A tire company has developed a new type of steel-belted...

If a random sample of 12 of the new type of tire are tested, what is the probability that at least 2 of them will last longer than 75000km?

A tire company has developed a new type of steel-belted radial tire. Extensive testing indicates that the population of lifetimes obtained by all tires of this new type is normally distributed with a mean of 65000km and a standard deviation of 6500km.

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### 1 Answer

The population of tires that are tested is normally distributed with a mean of 65000 km and a standard deviation of 6500 km.

A random sample of 12 tires is tested and the probability that at least 2 of them will last longer than 75000 km has to be determined. The probability that at least 2 of them will last longer than 75000 km is equal to 1 - (probability than none will last 75000 km) - (probability that 1 will last 75000 km)

Using a normal probability distribution table, the z-score of a tire lasting 75000 km is (75000 - 65000)/6500 = 1.538

The probability for a score greater than 1.538 is 0.7019. Of the 12 tires, the probability that any tire lasts greater than 75000 km is 0.7019. The probability that it does not last greater than 75000 km is 0.2981. The probability that at least two tires last greater than 75000 km is 1 - 12*0.2981*(0.7019)^11 - 0.2981^12 = 0.9271

**The probability that 2 of 12 tires picked at random lasts more than 75000 km is 0.9271.**

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