Bitcoin instant transactions are less secure

[o_e_l_e_o]

Furthermore (if I remember my maths correctly, but correct me if I'm wrong on this one) your calculation isn't ONLY for the probability of finding a block within 2 minutes of the previous block. It's also the probability that a block will be found in the NEXT 2 MINUTES, no matter how long you've already been waiting.  So, after 10 minutes of waiting since the most recently found block, there's still an 18.1% probability that the next block will show up in the NEXT 2 MINUTES. After an hour of waiting since the most recently found block, there's still an 18.1% probability that the next block will show up in the NEXT 2 MINUTES.

Everything you have written is correct, but I wanted to expand on this last point. The probability of the next 2 minutes is always the same, but the probability of a specific 2 minutes changes.

For example. Let's say I am interested in a block being mined between minutes 10 and 12. I calculate the probability of mining a block in the next 12 minutes, and I calculate the probability of mining a block in the next 10 minutes, and I subtract the two. The probability, then, of finding a block between minutes 10 and 12 works out to 6.7%.

However, by the time we get to the 10 minute mark, that probability is now 18.1%. Obviously this makes sense when you think about it, as every 2 minute bracket can't have a probability of 18.1%, or else we would rapidly end up with >100% chance.

Bitcoin mining is a memoryless process. It doesn't matter what came before - it is always 10 minutes on average until the next block.

[lixer]

As you know, each Bitcoin transaction usually takes a few seconds, and confirmation of the same transaction starts ten minutes after that. During this time, interaction is permitted and may also be reversible. Deceptive users try to cheat. If you can't wait for approval, request a small transaction fee or use the Unsafe Interactions Detection System, which can enhance security. For higher amounts, e.g. US$1,000, it makes sense to wait for 6 confirmations or more. Any confirmation can reduce the risk of reverse transactions exponentially.

The confirmations are very important for a reason, but I have never seen a bitcoin transaction reversed before. Maybe there are big companies, that runs wallet and owns a mining farm that may be able to do that, but I have not seen or heard that it was reversed before. That aside, the transaction fees that you’re paying for the transactions you’re making with bitcoin are what determines how fast your transaction is going to be. When you’re doing this transaction you have three options and the highest option there which is priority tries to make sure that your transaction is in the next block and gets confirmed immediately.

[RapTarX]

The confirmations are very important for a reason, but I have never seen a bitcoin transaction reversed before. Maybe there are big companies, that runs wallet and owns a mining farm that may be able to do that, but I have not seen or heard that it was reversed before. That aside, the transaction fees that you’re paying for the transactions you’re making with bitcoin are what determines how fast your transaction is going to be. When you’re doing this transaction you have three options and the highest option there which is priority tries to make sure that your transaction is in the next block and gets confirmed immediately.

There's no reverse actually but there's double spend in which you can use the same input for creating two transactions where the later one (likely) will be confirmed as you are going to pay a higher fee for that transaction. Thus, the prior one is invalid and you can successfully have spent the same input twice. Well, 1 confirmation is more than enough for even $100k now I guess as the network is quite more mature now.
Which fee options are you talking about? I guess you are talking about blockchain wallet which is quite misleading in terms of the actual fee model. You generally pay fee per byte.

[Shenzou]

As you know, each Bitcoin transaction usually takes a few seconds, and confirmation of the same transaction starts ten minutes after that. During this time, interaction is permitted and may also be reversible. Deceptive users try to cheat. If you can't wait for approval, request a small transaction fee or use the Unsafe Interactions Detection System, which can enhance security. For higher amounts, e.g. US$1,000, it makes sense to wait for 6 confirmations or more. Any confirmation can reduce the risk of reverse transactions exponentially.

I think that is one of the biggest problems that we re are facing with bitcoin, lets say you are a shop owner who is trying to make bitcoin a payment system, how do you go about accepting transactions, for goods do you wait for confirmations before giving the goods, do you demand you customers to pay higher transaction fees in order to speed up the transaction, or do you personally cover it, i feel like there is more work need to make paying for day to day things more efficient.

[o_e_l_e_o]

Well, 1 confirmation is more than enough for even $100k now I guess as the network is quite more mature now.

The maturity of the network is irrelevant. There still exists a possibility of an attacker being able to reverse a 1 block deficit, even without 51% of the hashrate. Even an attacker with only 10% of the hash rate has a 20% chance of reversing a single confirmation. We still get stale blocks even without any active attacks on the network (although these rarely result in successful double spends, the possibility still exists). If $100k is small change to you, then sure, accept 1 confirmation. But if $100k is a significant amount of money to you, I'd be waiting for 3-6 confirmations.

I think that is one of the biggest problems that we re are facing with bitcoin, lets say you are a shop owner who is trying to make bitcoin a payment system, how do you go about accepting transactions, for goods do you wait for confirmations before giving the goods, do you demand you customers to pay higher transaction fees in order to speed up the transaction, or do you personally cover it

Credit card transactions do not result in money reaching the merchant for 3-5 working days, and can be reversed for 90-180 days after the transaction, and yet almost everywhere accepts credit card transactions because the majority of people are honest and will not try to scam. It is entirely reasonable for merchants to accept zero confirmations for transactions which are not that valuable, pay a reasonable fee, and are not opted in to RBF. If they are uncomfortable doing that, then they can open a Lightning channel.

[EvieLannister]

Instant transaction means that both parties will trust each other.
Without a third-party guarantee, if it is cheated, the chance of getting it back is not very high.
All transactions need to be careful.