One-step three-branch tree for an at-the-money call

A stock starts at $S_0 = 8$. Over one period it moves to one of three levels: up $30\%$ with probability $0.2$, unchanged with probability $0.5$, or down $50\%$ with probability $0.3$. Treat these as the pricing probabilities and ignore discounting. What is today's value of a European call struck at $K = 8$ that expires at the end of the period?

$0.48$. The terminal prices are $10.4$, $8$ and $4$; only the up state pays, giving $10.4 - 8 = 2.4$. Weighting by its probability gives $0.2 \times 2.4 = 0.48$. (A trinomial tree is incomplete, which is why the probabilities have to be supplied rather than derived.)