A plan to defeat a betting game where the odds of winning are 50/50. Help me understand why it's flawed. My friend has this plan where he implies that it's impossible to lose, as long as the odds of winning are 50/50 on each bet. His idea is that basically you keep doubling your bet until you win and then start over again.

So for example, you bet 1 dollar and you lose, your net profit is now -1 dollar. Now you double your bet to 2 dollars and you lose again so your net profit is -3 dollars. Now you double your bet to 4 dollars and you win. This means you gain 4 dollars and now your net profit is 1 dollar. So you've made a profit. Now you start again. The reasoning here being that it is highly unlikely for you to lose a 50/50 toss x number of times in a row.

My counter-argument here is that basically if you go in with 50 dollars with the aim of doubling up to 100 dollars, you have the same odds of winning if you do one bet of 50 dollars or the technique outlined above. I cannot wrap my head around explaining this issue in a clear manner though, so maybe you wonderful folk at Mathematics can help!

Oh and I've pointed out that he uses gamblers fallacy in very obscure way, as he insists you need to go back to betting 1 dollar once you've won. This appears to be an obscure case of gamblers fallacy to me as it implies there is some hidden force which are changing the odds on each individual coin toss.
Let's play out your specific scenario: you start with \$50 and want to stop at \$100.  That means you need to play your sequence (double until you win) 50 times without busting.

However, if your sequence begins with six losses, then you are busted, because you just lost $1+2+4+8+16+32=63$ dollars, and your system requires you to have \$64 left to bet.  Six losses in a row happens $1/2^6=\frac{1}{64}$ of the time.

To avoid 6 losses in a row, happens $\frac{63}{64}$ of the time, and you need to do this 50 times, so $(\frac{63}{64})^{50}\approx 0.455$.  Hence, if all you need to do is avoid six losses, there is about a 45% chance of "winning", i.e. getting to \$100.

However, there are other circumstances when you bust; at the beginning even five losses in a row will bust you, since $1+2+4+8+16=31$ dollars, and you need to bet \$32.  Hence until you get to \$63, you need to avoid five losses in a row.  Thus, the true answer is $$\left(\frac{31}{32}\right)^{13}\left(\frac{63}{64}\right)^{37}\approx 0.370$$
Thus your friend's system has a 37% chance of getting to \$100, and a 63% chance of getting to \$0.  You are much better off just betting the whole \$50 in a single bet.
The simple solution would be to sit and wait until 5 loses come. This will take statistically 32 turns. Then use this betting system (Assuming you lose) £5 £12 £30 £60 £70. At this point you have a 1/1024 chance of losing. £150 £200 £300 £500 £700. At this point you have a seriously low chance of this happening and when it does you'd be extremely likely to be rich already. You've risked £2027 however the chances that your previous winnings will cover this loss are 100.000s in your favour. Always stay with the same strategy of playing the game you're playing. For example: rollette. If you start on black stay on black. 

With your friends strategy you will always be playing to gain £1 profit. By his tenth loss he has put the bet on of 512 losing £1023 for that £1 profit WHAT THE HELL IS WRONG WITH HIS LOGIC. It gets worse by his 15th hand he would be placing £16384 and has already lost 16383 so when that fails he has lost 32767 all for chasing the £1 profit. This is just plain stupid. Casinos have minimums and maximum betings for a reason and that is to put them in favour to win. 

Now I'm not encouraging ANYONE to gamble but if you're going to do it regardless then please use mine over the original strategy. Casinos and so on are winners for a reason! Remember that. Bye for now and if you want to email me your thoughts please email andytomkins11@gmail.com p.s, my maths maybe slightly out as I done them in my head.