10.1 Die gebeurtenisse S en T is onafhanklik. - P(S en T) = \frac{1}{6} - P(S) = \frac{1}{4} 10.1.1 Bereken P(T). 10.1.2 Vervolgens, bereken P(S or T). 10.2 'n VYF-syfer-kode word uit die syfers 2 ; 3 ; 5 ; 7 ; 9 saamgestel. Hoeveel verskillende kodes kan saamgestel word as: 10.2.1 Herhaling van syfers NIE in die kode toegelaat word NIE 10.2.2 Herhaling van syfers WEL in die kode toegelaat word. 10.3 'n Groep van 3 Suid-Afrikaners, 2 Australiërs en 2 Engelse bly almal in dieselfde hotel tydens hul vakansie is. Elke persoon het sy/haar eie kamer en die 2 Australiërs se kamers direk langs mekaar sal wees en die 2 Engelse se kamers ook direk langs mekaar sal wees.

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10.1 Die gebeurtenisse S en T is onafhanklik - NSC Mathematics - Question 10 - 2017 - Paper 1

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10.1 Die gebeurtenisse S en T is onafhanklik. - P(S en T) = \frac{1}{6} - P(S) = \frac{1}{4} 10.1.1 Bereken P(T). 10.1.2 Vervolgens, bereken P(S or T). 10.2 'n VY... show full transcript

Worked Solution & Example Answer:10.1 Die gebeurtenisse S en T is onafhanklik - NSC Mathematics - Question 10 - 2017 - Paper 1

Step 1

Bereken P(T)

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Answer

Given that the events S and T are independent, we can use the formula for independent events:
$P(S \text{ and } T) = P(S) \times P(T)$
We know that: $P(S \text{ and } T) = \frac{1}{6}$
$P(S) = \frac{1}{4}$
Substituting the known values into the formula: $\frac{1}{6} = \frac{1}{4} \times P(T)$
To find P(T), we rearrange the equation:
$P(T) = \frac{1}{6} \div \frac{1}{4} = \frac{1}{6} \times \frac{4}{1} = \frac{4}{6} = \frac{2}{3}$
Thus,
$P(T) = \frac{2}{3}$

Step 2

Vervolgens, bereken P(S or T)

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Answer

Using the formula for the probability of the union of two independent events:
$P(S \text{ or } T) = P(S) + P(T) - P(S \text{ and } T)$
Substituting the known values: $P(S) = \frac{1}{4}, \ P(T) = \frac{2}{3}, \ P(S \text{ and } T) = \frac{1}{6}$
Thus: $P(S \text{ or } T) = \frac{1}{4} + \frac{2}{3} - \frac{1}{6}$
Finding a common denominator (which is 12):
$= \frac{3}{12} + \frac{8}{12} - \frac{2}{12} = \frac{9}{12} = \frac{3}{4}$
Therefore,
$P(S \text{ or } T) = \frac{3}{4}$

Step 3

Herhaling van syfers NIE in die kode toegelaat word NIE

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Answer

To determine the number of different codes using the digits 2, 3, 5, 7, and 9 without repetition, we can use the permutation formula:

$n(E) = n!$
Where n is the number of available digits. Here, n = 5 (because we have five digits).
Thus, the number of codes is: $n(E) = 5! = 120$

Step 4

Herhaling van syfers WEL in die kode toegelaat word

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Answer

If repetition is allowed, each of the five positions can be filled by any of the 5 digits.
Thus, the total number of different codes is:

$n(E) = 5^5 = 3125$

Step 5

Bepaal die waarskynlikheid dat die 2 Australiërs se kamers direk langs mekaar sal wees

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Answer

To compute the probability of the 2 Australian rooms being next to each other, consider the two Australians as a single unit or block.
Now, we have:

6 units in total (3 South Africans + 1 block of 2 Australians + 2 English)

The arrangements of these 6 units and the arrangement within the block: $n(E) = 6! \times 2!$
Total arrangements of all rooms without constraints: $n(S) = 7!$
Probability: $P(E) = \frac{n(E)}{n(S)} = \frac{6! \times 2!}{7!} = \frac{2}{21}$

10.1 Die gebeurtenisse S en T is onafhanklik - NSC Mathematics - Question 10 - 2017 - Paper 1

Worked Solution & Example Answer:10.1 Die gebeurtenisse S en T is onafhanklik - NSC Mathematics - Question 10 - 2017 - Paper 1

Bereken P(T)

Vervolgens, bereken P(S or T)

Herhaling van syfers NIE in die kode toegelaat word NIE

Herhaling van syfers WEL in die kode toegelaat word

Bepaal die waarskynlikheid dat die 2 Australiërs se kamers direk langs mekaar sal wees

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