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Reconnaître des situations relevant de la loi binomiale
Calculer une probabilité dans le cadre de la loi binomiale
Représenter la répétition d'expériences identiques et indépendantes à l'aide d'un arbre pondéré
Un schéma de Bernoulli d’ordre n et de paramètre p est la répétition d’une épreuve de Bernoulli de paramètre p n fois où
chaque issue est indépendante.
La loi de probabilité de la variable X comptant le nombre de succés de ce shéma de Bernouilli s'appelle la loi Binomiale et se note \mathcal{B}(n,p) .
Tirer un dé (faire 6 étant le succés) trois fois d'affilée est un schéma de Bernouilli d'ordre 3 et de paramètre \frac{1}{6} . On peu le représenter par
l'arbre pondéré suivant :
P (X=2) correspond aux chemins dans l'arbre : SS\bar{S} ou S\bar{S}S ou SS\bar{S} ou \bar{S}SS :
\begin{array}{lll}
P (X=2) &=& (\frac{1}{6}\times\frac{1}{6}\times\frac{5}{6}) + (\frac{1}{6}\times\frac{5}{6}\times\frac{1}{6}) + (\frac{5}{6}\times\frac{1}{6}\times\frac{1}{6}) \\\\
&=& \text{nombre de chemins a deux succes} \times (\frac{1}{6})^2\times (\frac{5}{6})^1
\end{array}
Le coefficient binomial \binom{n}{k} est le nombre de chemins réalisant k succés pour n répétitions.
Dans l'exemple précédent, le nombre de chemins à 2 succés parmi 3 tirs est \binom{2}{3} = 3
À la calculatrice, on peut calculer \binom{n}{k} de la manière suivante :
Texas Instrument : touche MATH
puis PRB
, puis Combinaison
. On tape n Combinaison k
Casio : touce OPTN
puis →
, puis PROB
puis nCr
. On tape n nCr k
La probabilité de faire k succés parmi n dans une épreuve de Bernouilli de paramètre p est :
P (X = k) = \binom{n}{k} p^k \times (1-p)^{n-k}
À la calculatrice, on peut calculer P (X=k) pour la loi binomiale\mathcal{B}(n,p) de la manière suivante :
Texas Instrument : Menu Distrib (2nde +var ) puis PRB
, puis BinomFdp
. On tape BinomFdp (n,p,k)
Casio : touche OPTN
puis STAT
, puis DIST
puis BINM
puis Bpd
. On tape BinomialPD (k, n, p)
À la calculatrice, on peut calculer P (X\leq k) pour la loi binomiale\mathcal{B}(n,p) de la manière suivante :
Texas Instrument : Menu Distrib (2nde +var ) puis PRB
, puis BinomFRép
. On tape BinomFRép (n,p,k)
Casio : touche OPTN
puis STAT
, puis DIST
puis BINM
puis Bcd
. On tape BinomialCD (k, n, p)
La probabilité de gagner 2 fois P (X=2)correspond aux chemins dans l'arbre : SS\bar{S} ou S\bar{S}S ou SS\bar{S} ou \bar{S}SS :
\begin{array}{lll}
P (X=2) &=& (\frac{1}{6}\times\frac{1}{6}\times\frac{5}{6}) + (\frac{1}{6}\times\frac{5}{6}\times\frac{1}{6}) + (\frac{5}{6}\times\frac{1}{6}\times\frac{1}{6}) \\\\
&=& \text{nombre de chemins a deux succes} \times (\frac{1}{6})^2\times (\frac{5}{6})^1
\end{array}