Question

A security alarm requires a four-digit code. The code can use the digits 0–9 and the digits cannot be repeated. Which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?

Answer 1

Answer:

$\text{required probability}=\frac{2\cdot 9\cdot 8\cdot 7}{\frac{10\cdot9\cdot 8\cdot 7\cdot 6!}{6!}}$

Step-by-step explanation:

We have total of 10 digits from 0 to 9

We need a four digit code and beginning with a number greater than 7

We have 2 choices for number greater than 7

For first digit we have 2 choices

Now, we are left with 9 choices because one is filled and digits should not repeat.

For second digit we have 9 choices

For third digit we will have 8 choices

And for fourth digit we will have 7 choices

Total number of choices are $^{10}P_4$

using $^{n}P_r=\frac{n!}{(n-r)!}$

$\text{required probability}=\frac{\text{favourable choices}}{\text{total choices}}$

$\text{required probability}=\frac{2\cdot 9\cdot 8\cdot 7}{\frac{10\cdot9\cdot 8\cdot 7\cdot 6!}{6!}}$

Cancel common terms from numerator and denominator we get:

$\text{required probability}=\frac{1}{5}$.

Answer 2

Probability is the chance of an event occurring. The probability of the alarm code beginning with a number greater than 7 is 20%.

What is the Probability?

The probability helps us to know the chances of an event occurring.

$\rm{Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}$

We know about the probability, therefore, in order to find the probability of the alarm code beginning with a number greater than 7. We need to find the number of possible choices of an alarm code beginning with a number greater than 7.

As it is given to us that the numbers can not repeat, therefore, the number of possible options between 0-9 reduces, also, in the first place the only numbers that can come is either 8 or 9 so that the code begins with a number greater than 7.

The number of possible options that the code begins with a number 7

$= ^2C_1 \times ^{9}C_1 \times ^{8}C_1 \times ^{7}C_1\\\\ = 1008$

The total number of possible codes

$=^{10}C_1 \times ^{9}C_1 \times ^{8}C_1 \times ^{7}C_1\\ = 5040$

Now, the probability of the alarm code beginning with a number greater than 7,

$\rm Probability= \dfrac{\text{Possible options that code begins with a number 7}}{\text{The total number of possible codes}}$

$\rm Probability= \dfrac{1008}{5040}\\\\Probability = 0.2 = 20\%$

Hence, the probability of the alarm code beginning with a number greater than 7 is 20%.

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