Is my probability formula correct? The Question

The response rate to telephone polls is around $2{\%}$. Of people who
respond to telephone polls, currently about $40{\%}$ approve of the
President. Suppose we perform a telephone poll by calling a random
sample of $50,000$ registered voters.
(a) Let $X$ be a binomial random variable representing the number of
people who respond to the telephone poll. What is the expected value
and standard deviation of $X$?
(b) Let $Y$ be a binomial random variable representing the number of
people who respond to the telephone poll and approve of the President.
What are the expected value and standard deviation of $Y$?
(c) Use R to find the probability that at least $420$ people respond
to the telephone poll and approve of the President. Give your R code.

My Understanding
This is my thought process:
For 1a, $X \sim \mathscr{B}(N, P)$, $N$ being the $50,000$ registered voters and $P$ being the $2\%$ (rate at which people are answering polls), so it would be $EV = 50000 * .02=1000$. The variance is $N * P *(1-P)$ so plugging in the numbers I would get $980$ and to get standard deviation it would just be the square-root of that so $31.3$
For 1b, it would be similar except replace $P$ with $40\% (.4)$, so the expected value would be $20,000$. The variance would be $50,000 * .4 * .6 = 12000$. Taking the square root would be $109.54$
Using R to find probability of at least $420$ people would just be dbinom(420, 50000, .02)
Please correct me if I am wrong, but I believe this is correct. Thank you in advance!
 A: The solution to 1a looks correct.
For 1b, you're right that its similar, but the proportion is not $40\%$. We're given that $Y$ represents the number of people who respond to the telephone poll and approve of the president. The probability that somebody responds and approves is
$$ P[\text{Respond} \cap \text{Approve}] = P[\text{Approve} | \text{Respond}]*P[\text{Respond}] $$
$$ = 0.40*0.02 $$
$$ = 0.008 . $$
So, we have $Y \sim \mathscr{B}(50000, 0.008)$. Then, the expected value is
$$ E[Y] = np = 50000*0.008 = 400 .$$
The variance is
$$ V[Y] = np(1-p) = 50000*.008*.992 = 396.8 , $$
which gives us a standard deviation of
$$ SD[Y] = \sqrt{V[Y]} = \sqrt{396.8} \approx 19.92 $$
For 1c, you don't want to use dbinom, it gives you the probability of a specific value of $Y$. If you used dbinom(420, 50000, .008), you'd end up calculating $P[Y = 420]$, but in this case, we want $P[Y \ge 420]$.
The more appropriate function in this case is pbinom (R documentation). Using that function, pbinom(k,n,p) results in the cumulative distribution up to and including the value of $k$. So, pbinom(k,50000,0.008) would be equal to $P[Y \le k]$. Then, since
$$ P[Y \ge 420] = 1 - P[Y < 420] = 1 - P[Y \le 419], $$
we have $k = 419$, and can calculate it with 1-pbinom(419,50000,0.008).
A: Answer Key
(a) $X\sim \mathscr B (50000,.02)$ so that $E(X)=50000(.02)=1000$, $\sigma_X=\sqrt{50000(.02)(.98)}=31.30495$. Yup yours looks right.
(b) You want $Y\sim\mathscr B(50000, .02*.4)=\mathscr B(50000,.008)$. It says that of the people who respond, 40% are in favor. So not 40% of the whole population is in favor. This is a conditional probability, but you don't really need to know the term to be able to multiply the two numbers. Repeating the calculation in (a), you get $E(Y)=50000(.008)=400$ and $\sigma_Y=\sqrt{50000*.008*.992}=19.91984$
(c) The correct commands are either of the following. Subtracting the lower tail from 1 is the same as getting the upper tail. You don't want to use dbinom, because if you do, you will have to type it hundreds of times, e.g. 1-[dbinom(0,50000,.008)+dbinom(1,50000,.008)+...+dbinom(419,50000,.008)].
> pbinom(419,50000,.008,lower.tail=FALSE)
[1] 0.1637013
> 1-pbinom(419,50000,.008,lower.tail=TRUE)
[1] 0.1637013

