There is a 7% chance that a band's lead singer's voice isn't working due to illness on any given Saturday. This band has played 19 Saturday gigs so far this year, and at each of these the lead singer's voice has been fine. What is the probability that the lead singer's voice will be fine at the band's next Saturday gig, which is 2 weeks from now? Assume she never is sick for more than 6 consecutive days, and the interval between bouts of illness is inherently unpredictable.

Answer:

Step-by-step explanation:

The probability that the lead singer's voice is fine on any given Saturday is 1 - 0.07 = 0.93. Since the lead singer's voice has been fine at each of the 19 Saturday gigs this year, the probability of this happening by chance is (0.93)^19 = 0.285, or 28.5%.

The fact that the lead singer's voice has been fine for 19 consecutive Saturday gigs does not guarantee that it will be fine at the next gig, but it does suggest that the singer is not currently suffering from a long-term illness that is affecting their voice.

Assuming that the lead singer's illnesses are independent and that they never last more than 6 consecutive days, we can use a Poisson process to model the probability of the lead singer's voice being fine at the next Saturday gig. Let X be the number of days since the last illness, then X is exponentially distributed with a mean of 1/0.93 = 1.0753 weeks.

The probability that the lead singer's voice will be fine at the next gig is equal to the probability that X is greater than or equal to 14 days (2 weeks). This probability can be calculated using the cumulative distribution function (CDF) of the exponential distribution:

P(X >= 14) = 1 - F(14)

 where F(x) = 1 - e^(-lambda*x) is the CDF of the exponential distribution with parameter lambda = 1/1.0753.

Using this formula, we get:

P(X >= 14) = 1 - F(14) = 1 - (1 - e^(-lambdax)) = e^(-lambdax) = e^(-1.0753*2) = 0.319

Therefore, the probability that the lead singer's voice will be fine at the band's next Saturday gig is approximately 31.9%.