Exercise 3 · Probability and optimisation — Tickets for a concert

Total and conditional probability with a tree, and the study of a cubic function to find when 100 decibels are exceeded.

Maximum score · 2.5 points

The Town Council of Canet de Mar has obtained 24 free tickets for a concert by a Catalan rock band and has decided to raffle them among the residents interested in attending. All the residents of the town who are fans of this band enter the raffle, and only 16% of them get a ticket. Of the remaining fans of the band, six sevenths try to buy a ticket through the website, where the probability of getting one is 25%.

How many people from this town have a ticket for the concert? 0.75 p
If a resident of Canet who is a fan of this rock band is chosen at random and does not have a ticket for the concert, what is the probability that they tried to get one through the website? 0.75 p
The decibel level during the song can be approximated by $S(t)=-t^3+12t^2-30t+90$, with $t\in[0,5]$ ($t$ in minutes). When 100 decibels are exceeded, some lighting effects are triggered. Will they be triggered at any moment during these five minutes? If the answer is yes, compute in which minute they are triggered, rounded to the tenths. 1 p

Science Bachillerato · Blocks E and D — Statistics and analysis Total probability Conditional probability Function on an interval

Step-by-step solution

Key idea: the raffle distributes exactly 24 tickets and corresponds to 16% of the fans, which fixes how many fans there are. From there, a tree sorts out who has a ticket and who does not. In part c) we must compare $S(t)$ with 100 over the whole interval $[0,5]$.

a) How many people have a ticket

Let $N$ be the number of fans. The raffle gives a ticket to 16% and that is exactly the 24 available tickets:

0{,}16\,N=24 \;\Longrightarrow\; N=\frac{24}{0{,}16}=150\ \text{fans.}

Tickets from the raffle: $24$. Without a raffle ticket, $150-24=126$ remain. Of these, six sevenths try the website:

\tfrac{6}{7}\cdot 126=108 \text{ try}, \qquad 25\%\text{ succeed}: \;0{,}25\cdot 108=27.

Total with a ticket $=24+27=51$.

There are $150$ fans; $51$ people have a ticket ($24$ from the raffle $+\,27$ from the website).

b) Probability of having tried the website, given no ticket

We count those without a ticket ($150-51=99$) and, among them, those who had tried the website:

Tried the website and did not get one: $108-27=81$.
Did not try the website (nor had one): $126-108=18$.

In total $81+18=99$ without a ticket, as expected. The conditional probability is:

P(\text{web}\mid\text{no ticket})=\frac{81}{99}=\frac{9}{11}\approx 0{,}818.

$P=\dfrac{9}{11}\approx 0{,}818$ (approximately 81.8%).

c) Are the lighting effects triggered?

We need to know whether $S(t)$ exceeds $100$ on $[0,5]$. We study the extrema with the derivative:

S'(t)=-3t^2+24t-30=-3\,(t^2-8t+10)=0 \;\Longrightarrow\; t=4\pm\sqrt{6}.

Within $[0,5]$ there is only $t=4-\sqrt{6}\approx 1{,}55$. Since $S''(t)=-6t+24>0$ here, it is a minimum. Therefore the maximum of $S$ on the interval is attained at an endpoint. We evaluate:

S(0)=90,\qquad S(5)=-125+300-150+90=115.

Since $S(5)=115>100$, the effects are triggered. We find the moment when $S(t)=100$:

-t^3+12t^2-30t+90=100 \;\Longrightarrow\; t^3-12t^2+30t+10=0.

The function decreases from $S(0)=90$ to the minimum $S(1{,}55)\approx 68{,}6$ and then rises to $S(5)=115$, so it crosses the level 100 only once, on the increasing part. Solving numerically:

t\approx 4{,}11 \;\text{min} \;\Rightarrow\; t\approx 4{,}1 \text{ minutes.}

Yes, they are triggered: the level exceeds 100 dB from $t\approx 4{,}1$ minutes on (and stays there until the end, $t=5$).