A company is going to launch an exclusive high-end smartphone. They rely on a bold marketing strategy that offers a lifetime warranty for replacing the first failure caused by normal use in each smartphone sold.

The engineering team does a precise accelerated aging test on 106 smartphones and detects that 5 fail. The test is precise in simulating the real conditions of a lifetime of use.

A 0.0472 probability of failure is just within the long-run budget for the marketing campaign and everyone in the company is in good spirits, until someone in the analytics department warns them that the probability of failure for each smartphone could be much higher. For example, if the true probability of failure was 0.0566 the budget would be greatly exceeded (by 20% !) and the company would have to close. What is the probability of having observed the results of the aging test, or even better results (less failures), if the true probability of failure is 0.0566

Please use the normal approximation and introduce the answer to 3 decimal places.