Adding multiple percentages to value

percentages

Good morning,

Apologies if this question seems obvious but I need a little help with adding multiple percentages to a value.

The value: $590,568$
The percentages: $14.62%, 49.068%, 2%$

My understanding is as follows:

$\frac{14.62 }{ 100 }= 0.1462$

$590,568 \times 0.1462 = 86,341.0416$

$\frac{49.068 }{ 100 }= 0.49068$

$590,568 \times 0.49068 = 289,779.90624$

$\frac{2 }{100} = 0.02$

$590,568 \times 0.02 = 11,811.36$

$590,568 + 86,341.0416 + 289,779.90624 + 11,811.36 = 387,932.30784 $

$590,568 + 387,932.30784 = 978,500.30784$

My question is if this is correct or should I be multiplying each percentage (after it's been divided by $100$) by the new total so e.g. ($590,568 + 86,341.0416) \times 49.068$

Thank you in advance

Best Answer

You did well.

You can also think of it like this: all of those percentages are going to be portions of the same quantity (590,568), so you can add them together first $$14.62\% + 49.068\% + 2\% = 65.688\%$$

then you take the 65.688% of 590,568 (which is 590,568*0.65688 = 387,932.3) and that's the amount that you need to add: $$590,568 + 387,932.3 = 978,500.3$$

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[Math] Calculating Percentages/Probabilities of a Specific Scenario.

I'll post this answer assuming that it's possible to win no prize, which seems reasonable.

Of course, there's no unique answer to this question, since for example, you could simply always give back the $8^{th}$ prize. Still, I took an approach that tries to go around this problem of non-uniqueness.

Let's use for that the concept of RTP. We can force the game to make every prize has the same RTP and that all together the game has an RTP of 100%. What I mean by every prize having the same RTP is that each price, taking into account how often it appears, contribute the same to the total RTP. So for each prize the formula $prize\times P(prize)$ remains constant. That means that $RTP=4\%$.

Now let's find what should be $P(prize_n)$ where $prize_n$ is winning the $n^{th}$ prize.

$$P(prize_n)=2\times\frac{4\%}{prize\ value} $$

This gives the following table of probabilities

Type Value Probability Frecuency

 1.  $0.5  0.16        6.25
 2.  $0.5  0.16        6.25
 3.  $1    0.08        12.5
 4.  $1    0.08        12.5
 5.  $1    0.08        12.5
 6.  $1    0.08        12.5
 7.  $2    0.04        25
 8.  $2    0.04        25
 9.  $2    0.04        25
 10. $4    0.02        50
 11. $5    0.016       62.5
 12. $5    0.016       62.5
 13. $7    0.01143*    87.5
 14. $8    0.01        100
 15. $10   0.008       125
 16. $10   0.008       125
 17. $12   0.00667*    150
 18. $15   0.00533*    187.5
 19. $15   0.00533*    187.5
 20. $20   0.004       250
 21. $20   0.004       250
 22. $20   0.004       250
 23. $25   0.0032      312.5
 24. $30   0.00267*    375
 25. $100  0.0008      1250

Probabilities with * had been rounded so if you have to implement it, I'd recommend you to use frecuencies ($\frac{1}{probability}$), that are not rounded. That's why I added the Frecuency column as well. Frecuencies are telling you that you should see that prize once for every time you'd played that number of games. So, you should see one prize 7 every 25 prize (that's the expected value), or after 100 games you should see 16 prizes 1.

The solution for your problem if you always win a prize requires some computation.

Let $p_i$ be the probability of wining prize $i$. Let $v_i$ the value of the przie $i$. You have the following system.

$$ \sum_{i=1}^{25} p_i=1\\ \sum_{i=1}^{25} p_iv_i=2\\ p_i>p_{i+1}\ \ \forall i\in [1,24] $$

Last condition wouldn't be needed but it's for making it more beautiful, every prize is more probable the smaller it is.

Here you have 26 equations (actually 2 equations and 24 inequations) that can solve through computer.

Appropriate Mathematical Symbol for Percentage Representation: $=$ or $\implies$

When representing percentages, it is common to use the equal sign (=) to indicate equivalence between the percentage value and its decimal form. $80\%$ can be represented as $0.8$. This is because percentages are simply a way of expressing a proportion or ratio out of 100. So, saying "$p = 80\%$" is equivalent to saying "$p = 0.8$". Both statements convey the same information.

The implication symbol ($\implies$) is used to indicates a logical implication or deduction. It denotes that one statement logically implies another.

In the case of representing percentages, the implication symbol is not commonly used because there is no logical implication between the percentage value and its decimal form. It might even cause confusion. Instead, the equal sign ($=$) is used to show the equivalence.

To conclude, it is generally acceptable to use the equal sign ($=$) to represent the equivalence between a percentage value and its decimal form. So, you can write "$p = 80\% = 0.8$". Using the implication symbol ($\implies$) would not be appropriate in this context.

Best Answer

Related Solutions

[Math] Calculating Percentages/Probabilities of a Specific Scenario.

Appropriate Mathematical Symbol for Percentage Representation: $=$ or $\implies$

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