Tutor profile: Jonathan K.

Find the last digit of $2019^2019$

9. The key observation here is that the last digit of $9x$ depends only on the last digit of $x$. In particular in $x$ ends in a $9$ then $9x$ ends in a $1$ and if $x$ ends in a $1$ then $9x$ ends in a $9$. From here it is easy to see that the last digit of a power of $2019$ will alternate between $9$ and $1$, with even powers ending in $1$ and odd powers (such as $2019^2019$ ending in $9$

A fair coin is flipped 300 times. Find the probability of obtaining an even number of heads.

0.5. Consider the first 299 flips. If we end up with an even number of heads after 299 flips we'd need a tail on flip number 300 to end on an even number of flips. On the other hand if we have an odd number of heads after 299 flips, we need a head on flip 300 to end on an even number of flips. In either case we have a 50% chance, so the overall probability of an even number of heads is exactly 0.5.

A coin is flipped 10 times. We wish to test the null hypothesis that the coin is fair against the alternate hypothesis that the coin is more likely to turn up heads than tails at the 5% level. How many of the 10 flips would need to be heads before we'd reject the null hypothesis.

We'd require 9 heads. Letting X be the number of heads we compute: P(X=10)=1/1024 P(X=9)=10/1024 P(X=8)=45/1024. From this it follows that P(X>=9)=11/1024=0.01074219<0.05 and P(X>=8)=56/1024=0.0546875>0.05. So we would reject the null hypothesis of a fair coin if we saw 9/10 heads but would not reject the null hypothesis if we saw 8/10 heads.