Probability of loss using a binary symmetric channel
R3 encoding is a pair of algorithms:

Encoding: a bit $b \in \{0,1\}$ is encoded as the packet $bbb$.
Decoding: a packet $xyz \in \{0,1\}^3$ is decoded as the majority bit (the bit occurring most often).

Since the situation is symmetric, in order to understand the transmission error, it suffices to consider the case $b = 0$. Suppose we encode the bit $0$. So we transmit the packet $000$. Every bit is flipped with probability $p$, and this gives us a distribution on the received packet $xyz$. Finally, a transmission error occurs if $xyz$ is decoded as $1$, that is, if the majority of $xyz$ is $1$.
Therefore, the transmission error is the probability of the following event:

If $x,y,z \in \{0,1\}$ are independent identically distributed bits such that $\Pr[x=1] = p$, what is the probability that the majority of bits in $x,y,z$ is $1$?

You can compute this probability by brute force.