Final answer:
To find the probability that less than 81% of the sample diamonds failed to qualify as gemstone grade, we first calculate the probability that 81% or more of the sample diamonds failed.
Step-by-step explanation:
To find the probability that less than 81% of the sample diamonds failed to qualify as gemstone grade, we first need to find the probability that 81% or more of the sample diamonds failed to qualify. Since 75% of the diamonds failed to qualify as gemstone grade in the mine.
The probability that a randomly selected diamond fails to qualify is 0.75. Therefore, the probability that a randomly selected diamond qualifies as a gemstone is 1 - 0.75 = 0.25. To find the probability that 81% or more of the sample diamonds failed to qualify, we need to find the cumulative probability of getting 89 or more diamonds that fail to qualify.
Finally, to find the probability that less than 81% of the sample diamonds failed to qualify, we subtract the previous probability from 1: 1 - 0.1085 = 0.8915. Therefore, the probability that less than 81% of the sample diamonds failed to qualify as gemstone grade is approximately 0.8915, or 89.15%.