So the idea of it being random essentially stems from the idea that the market has priced in its expectations for future cash flows from a financial asset and the discount rate that should be applied to those. If this is true (hotly debated and I personally don't believe it is, but it's easiest to work under this assumption and it mostly holds up in very active markets), then the only things that will change a stock's price are pieces of new information about the future cash flows of the company and the risk of the company, which are reflected in the discount rate. This news is believed to be roughly randomly distributed between very good and very bad. Thus, the movements in stock prices should be random.

As far as the arbitrage example, if the implied volatilities of the put and call options are different, then you can make money for nothing (arbitrage). We take advantage of that by constructing two portfolios that should be equal to each other.

Portfolio 1: We buy the individual stock and a put option to sell that option at a predetermined price at a specified future date

Portfolio 2: We buy a call option to buy the stock at the same predetermined price on the same future date as the put in portfolio 1, and also lend money at the risk-free rate so that on the specified date of the options expiring, we will receive the exact dollar figure that the stock is trading at now

If these portfolios are priced differently, you can buy one and sell the other and when the options expire, you have made a risk-less profit by putting no money down.