The easiest approach is to let $\theta = P(\mathrm{Left\; Lat)},$ ignoring the distinction between Right and Bi-lateralization. Then find the number $x$ with Left lateralization
in your sample of $n = 38.$

**With an uninformative prior** such as $\theta \sim \mathsf{Beta}(.5,.5),$ a Jeffreys prior distribution, and with data $x = 30,$ your posterior distribution on $\theta$ would be
$\mathsf{Beta}(30.5, 8.5).$ Then a Bayesian posterior estimate of $\theta$ is
$E(\theta) = \frac{30.5}{30.5+8.5} = 78.2.$ Also, a 95% Bayesian posterior interval estimate of $\theta$ ('credible interval') would be $(0.641,0.895),$ as computed in R below.

    qbeta(c(.025,.975), 30.5, 8.5)
    [1] 0.6416816 0.8951003

In that case, your prior distribution has very little effect on the result. Numerically, the Bayesian
interval estimate is not substantially different from a frequentist 95% confidence interval based on $30$ binomial 'successes' in $n = 38$ trials.

**An informative prior** $\theta \sim \mathsf{Beta}(200, 300)$ would be
consistent with belief that $\theta \approx 0.6$ and $P(0.36 \le \theta \le 0.44) \approx 0.95.$

    qbeta(c(.025,.975), 200, 300)
    [1] 0.3575005 0.4432573

With this prior and again data $x=30$ out of $n=38$ subjects, the posterior
distribution is $\mathsf{Beta}(230, 308),$ the posterior mean is $44.2$ and a 95% Bayesian probability interval is $(0.386,0,470).$ 
In this case, the prior overwhelms the data, so that the result is mainly due to the prior. 

    qbeta(c(.025,.975), 230, 308)
    [1] 0.3860167 0.4695124

How strong the prior distribution should be would depend upon how sure you are about $\theta$ after taking your past experience and your literature review into account. A productive approach is to seek an equitable balance between information from the prior and information from the data.

**Bayes' Theorem.** Both of the results above depend on using Bayes' Theorem to multiply the kernel of the beta prior (density function without norming constant) by the binomial likelihood to get eh kernel of the beta posterior distribution.